Curs 5
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Groups(sequel)
The isomorphism theorems
M.Chi¸s ()
Lecture 5
3.XI.2008
1 / 26
Groups(sequel)
The isomorphism theorems Proposition (The fundamental isomorphism theorem) If f : (G , ·) −→ (T , ·) is a group homomorphism, then there is an unique isomorphism f : G /Ker (f ) −→ Im(f ) , such that πKer (f ) · f · iIm(f ) = f .
M.Chi¸s ()
Lecture 5
3.XI.2008
1 / 26
Groups(sequel)
The isomorphism theorems Proposition (The fundamental isomorphism theorem) If f : (G , ·) −→ (T , ·) is a group homomorphism, then there is an unique isomorphism f : G /Ker (f ) −→ Im(f ) , such that πKer (f ) · f · iIm(f ) = f .
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Remark The isomorphism f : G /Ker (f ) −→ Im(f ) is the unique application which makes the following diagram commutative: G π↓
f
−−−−→
T ↑i
f
G /Ker (f ) −−−−→ Im(f )
Corollary If f : (G , ·) −→ (T , ·) is a group homomorphism, then G /Ker (f ) ∼ = Im(f ) .
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Remark The isomorphism f : G /Ker (f ) −→ Im(f ) is the unique application which makes the following diagram commutative: G π↓
f
−−−−→
T ↑i
f
G /Ker (f ) −−−−→ Im(f )
Corollary If f : (G , ·) −→ (T , ·) is a group homomorphism, then G /Ker (f ) ∼ = Im(f ) .
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Proposition (The first isomorphism theorem) If f : (G , ·) −→ (T , ·) is a surjective group homomorphism, H E G with Ker (f ) ⊆ H, and U = (H)f , then G /H ∼ = T /U.
Proposition (The second isomorphism theorem) Let (G , ·) be a group, H ≤ G and N E G . Then H ∩ N E H and H/(H ∩ N) ∼ = HN/N .
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Proposition (The first isomorphism theorem) If f : (G , ·) −→ (T , ·) is a surjective group homomorphism, H E G with Ker (f ) ⊆ H, and U = (H)f , then G /H ∼ = T /U.
Proposition (The second isomorphism theorem) Let (G , ·) be a group, H ≤ G and N E G . Then H ∩ N E H and H/(H ∩ N) ∼ = HN/N .
Proposition (The third isomorphism theorem) Let (G , ·) be a group and K , N E G , with K ⊆ N. Then N/K E G /K and (G /K )/(N/K ) ∼ = G /N . M.Chi¸s ()
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Proposition (The first isomorphism theorem) If f : (G , ·) −→ (T , ·) is a surjective group homomorphism, H E G with Ker (f ) ⊆ H, and U = (H)f , then G /H ∼ = T /U.
Proposition (The second isomorphism theorem) Let (G , ·) be a group, H ≤ G and N E G . Then H ∩ N E H and H/(H ∩ N) ∼ = HN/N .
Proposition (The third isomorphism theorem) Let (G , ·) be a group and K , N E G , with K ⊆ N. Then N/K E G /K and (G /K )/(N/K ) ∼ = G /N . M.Chi¸s ()
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Proposition (The corespondence theorem for groups) Let (G , ·) be a group, K E G and πK : G −→ G /K the canonical projection. The function πK establishes then a bijective corespondence M ←→ M ∗ between the set of subgroups of G which include K and the set of all subgroups of G /K . Also, for any L, M ≤ G with K ⊆ L, M we have
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Proposition (The corespondence theorem for groups) Let (G , ·) be a group, K E G and πK : G −→ G /K the canonical projection. The function πK establishes then a bijective corespondence M ←→ M ∗ between the set of subgroups of G which include K and the set of all subgroups of G /K . Also, for any L, M ≤ G with K ⊆ L, M we have 1) M ∗ = M/K = (M)πK .
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Proposition (The corespondence theorem for groups) Let (G , ·) be a group, K E G and πK : G −→ G /K the canonical projection. The function πK establishes then a bijective corespondence M ←→ M ∗ between the set of subgroups of G which include K and the set of all subgroups of G /K . Also, for any L, M ≤ G with K ⊆ L, M we have 1) M ∗ = M/K = (M)πK . 2) L ⊆ M ⇐⇒ L∗ ⊆ M ∗ , and in this case [M : L] = [M ∗ : L∗ ].
M.Chi¸s ()
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Proposition (The corespondence theorem for groups) Let (G , ·) be a group, K E G and πK : G −→ G /K the canonical projection. The function πK establishes then a bijective corespondence M ←→ M ∗ between the set of subgroups of G which include K and the set of all subgroups of G /K . Also, for any L, M ≤ G with K ⊆ L, M we have 1) M ∗ = M/K = (M)πK . 2) L ⊆ M ⇐⇒ L∗ ⊆ M ∗ , and in this case [M : L] = [M ∗ : L∗ ]. 3) L E M ⇐⇒ L∗ E M ∗ , and in this case M/L ∼ = M ∗ /L∗ .
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Proposition (The corespondence theorem for groups) Let (G , ·) be a group, K E G and πK : G −→ G /K the canonical projection. The function πK establishes then a bijective corespondence M ←→ M ∗ between the set of subgroups of G which include K and the set of all subgroups of G /K . Also, for any L, M ≤ G with K ⊆ L, M we have 1) M ∗ = M/K = (M)πK . 2) L ⊆ M ⇐⇒ L∗ ⊆ M ∗ , and in this case [M : L] = [M ∗ : L∗ ]. 3) L E M ⇐⇒ L∗ E M ∗ , and in this case M/L ∼ = M ∗ /L∗ .
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Cyclic groups
Proposition Let (Z, +) be the additive group of the integers, and H ≤ Z. Then there is a nonnegative integer n ∈ N such that H = nZ.
Remark Let n ∈ N∗ be a positive integer. Then the binary relations on Z of congruence modulo n, respectively modulo the subgroup nZ, coincide: k ≡ l(mod n) ⇐⇒ n|k − l ⇐⇒ k − l ∈ nZ ⇐⇒ k ≡ l(mod nZ) . Hence, Zn := Z/≡(mod n) = Z/nZ.
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Cyclic groups
Proposition Let (Z, +) be the additive group of the integers, and H ≤ Z. Then there is a nonnegative integer n ∈ N such that H = nZ.
Remark Let n ∈ N∗ be a positive integer. Then the binary relations on Z of congruence modulo n, respectively modulo the subgroup nZ, coincide: k ≡ l(mod n) ⇐⇒ n|k − l ⇐⇒ k − l ∈ nZ ⇐⇒ k ≡ l(mod nZ) . Hence, Zn := Z/≡(mod n) = Z/nZ.
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Proposition Let (G , ·) be a cyclic group. Then either G ∼ = Z, or there is a positive integer n ∈ N∗ such that G ∼ Z . = n
Proposition Let (G , ·) be a finite cyclic group, of order |G | = n.
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Proposition Let (G , ·) be a cyclic group. Then either G ∼ = Z, or there is a positive integer n ∈ N∗ such that G ∼ Z . = n
Proposition Let (G , ·) be a finite cyclic group, of order |G | = n. 1) If H ≤ G , then there is a positive integer d ∈ N∗ , with d|n, such that H∼ = db · Zn .
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Proposition Let (G , ·) be a cyclic group. Then either G ∼ = Z, or there is a positive integer n ∈ N∗ such that G ∼ Z . = n
Proposition Let (G , ·) be a finite cyclic group, of order |G | = n. 1) If H ≤ G , then there is a positive integer d ∈ N∗ , with d|n, such that H∼ = db · Zn . 2) For any divisor d ∈ N∗ of the order n of the group, G has a unique subgroup of order d.
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Proposition Let (G , ·) be a cyclic group. Then either G ∼ = Z, or there is a positive integer n ∈ N∗ such that G ∼ Z . = n
Proposition Let (G , ·) be a finite cyclic group, of order |G | = n. 1) If H ≤ G , then there is a positive integer d ∈ N∗ , with d|n, such that H∼ = db · Zn . 2) For any divisor d ∈ N∗ of the order n of the group, G has a unique subgroup of order d. 3) The number of elements a ∈ G such that G = hai is (n)ϕ , where ϕ is Euler’s function(given by (n)ϕ =the number of nonnegative integers k, with 0 ≤ k ≤ n − 1, such that (k, n) = 1).
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Proposition Let (G , ·) be a cyclic group. Then either G ∼ = Z, or there is a positive integer n ∈ N∗ such that G ∼ Z . = n
Proposition Let (G , ·) be a finite cyclic group, of order |G | = n. 1) If H ≤ G , then there is a positive integer d ∈ N∗ , with d|n, such that H∼ = db · Zn . 2) For any divisor d ∈ N∗ of the order n of the group, G has a unique subgroup of order d. 3) The number of elements a ∈ G such that G = hai is (n)ϕ , where ϕ is Euler’s function(given by (n)ϕ =the number of nonnegative integers k, with P0 ≤ k ≤ n − 1, such that (k, n) = 1). 4) (d)ϕ = n. d|n
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Proposition Let (G , ·) be a cyclic group. Then either G ∼ = Z, or there is a positive integer n ∈ N∗ such that G ∼ Z . = n
Proposition Let (G , ·) be a finite cyclic group, of order |G | = n. 1) If H ≤ G , then there is a positive integer d ∈ N∗ , with d|n, such that H∼ = db · Zn . 2) For any divisor d ∈ N∗ of the order n of the group, G has a unique subgroup of order d. 3) The number of elements a ∈ G such that G = hai is (n)ϕ , where ϕ is Euler’s function(given by (n)ϕ =the number of nonnegative integers k, with P0 ≤ k ≤ n − 1, such that (k, n) = 1). 4) (d)ϕ = n. d|n
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Direct and semidirect products of groups
Direct products
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Direct and semidirect products of groups
Direct products Definition Let (G , ·) be a group and H, K E G . G is called the internal direct product of its normal subgroups H and K , if G = HK and H ∩ K = 1.
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Direct and semidirect products of groups
Direct products Definition Let (G , ·) be a group and H, K E G . G is called the internal direct product of its normal subgroups H and K , if G = HK and H ∩ K = 1.
Proposition If (G , ·) is the direct product of its normal subgroups H and K , then hk = kh, (∀)h ∈ H, k ∈ K and for any g ∈ G there are unique elements h ∈ H and k ∈ K such that g = hk.
M.Chi¸s ()
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Direct and semidirect products of groups
Direct products Definition Let (G , ·) be a group and H, K E G . G is called the internal direct product of its normal subgroups H and K , if G = HK and H ∩ K = 1.
Proposition If (G , ·) is the direct product of its normal subgroups H and K , then hk = kh, (∀)h ∈ H, k ∈ K and for any g ∈ G there are unique elements h ∈ H and k ∈ K such that g = hk.
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Remark If (G , ·) is the direct product of its normal subgroups H and K , and g1 = h1 k1 , g2 = h2 = k2 , with h1 , h2 ∈ H, k1 , k2 ∈ K , then g1 g2 = (h1 k1 )(h2 k2 ) = h1 (k1 h1 )k2 = h1 (h2 k1 )k2 = (h1 h2 )(k1 k2 ) .
Proposition Let (H, ·) and (K , ·) be two groups. Then the binary operation defined on the cartesian product H × K by (h1 , k1 ) · (h2 , k2 ) = (h1 h2 , k1 k2 ) defines on H × K a structure of a group.
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Remark If (G , ·) is the direct product of its normal subgroups H and K , and g1 = h1 k1 , g2 = h2 = k2 , with h1 , h2 ∈ H, k1 , k2 ∈ K , then g1 g2 = (h1 k1 )(h2 k2 ) = h1 (k1 h1 )k2 = h1 (h2 k1 )k2 = (h1 h2 )(k1 k2 ) .
Proposition Let (H, ·) and (K , ·) be two groups. Then the binary operation defined on the cartesian product H × K by (h1 , k1 ) · (h2 , k2 ) = (h1 h2 , k1 k2 ) defines on H × K a structure of a group.
Definition Let (H, ·) and (K , ·) be two groups. The group defined in the previous proposition is called the external direct product of the groups H and K . M.Chi¸s ()
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Remark If (G , ·) is the direct product of its normal subgroups H and K , and g1 = h1 k1 , g2 = h2 = k2 , with h1 , h2 ∈ H, k1 , k2 ∈ K , then g1 g2 = (h1 k1 )(h2 k2 ) = h1 (k1 h1 )k2 = h1 (h2 k1 )k2 = (h1 h2 )(k1 k2 ) .
Proposition Let (H, ·) and (K , ·) be two groups. Then the binary operation defined on the cartesian product H × K by (h1 , k1 ) · (h2 , k2 ) = (h1 h2 , k1 k2 ) defines on H × K a structure of a group.
Definition Let (H, ·) and (K , ·) be two groups. The group defined in the previous proposition is called the external direct product of the groups H and K . M.Chi¸s ()
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Proposition Let (H × K , ·) be the external direct product of the groups (H, ·) and b K b ⊆ H × K the subsets of the cartesian product H × K (K , ·), and H, defined by b := H × {1K } = {(h, 1K )| h ∈ H} , H b := {1H } × K = {(1H , k)| k ∈ K } . K b K b E H × K, H ∼ b K∼ b, H × K = H bK b and H b ∩K b = 1. Then H, = H, =K
Remark According to the previous proposition, the external direct product b and K b H × K is the internal direct product of the isomorphic copies H of the groups H and K . Consequently, we can identify the notions of internal and external direct product. If a group (G , ·) is the internal direct product of its subgroups H, K E G , we write G = H × K and we shall simply call G the direct product of H and K .
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Proposition Let (H × K , ·) be the external direct product of the groups (H, ·) and b K b ⊆ H × K the subsets of the cartesian product H × K (K , ·), and H, defined by b := H × {1K } = {(h, 1K )| h ∈ H} , H b := {1H } × K = {(1H , k)| k ∈ K } . K b K b E H × K, H ∼ b K∼ b, H × K = H bK b and H b ∩K b = 1. Then H, = H, =K
Remark According to the previous proposition, the external direct product b and K b H × K is the internal direct product of the isomorphic copies H of the groups H and K . Consequently, we can identify the notions of internal and external direct product. If a group (G , ·) is the internal direct product of its subgroups H, K E G , we write G = H × K and we shall simply call G the direct product of H and K .
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We shall generalize the notion of a direct product to arbitrary families of groups in the following way:
Definition If {Hi }i∈I is a family Q of groups, its direct product, denoted ×i∈I Hi is the cartesian product Hi , endorsed with the binary operation defined by i∈I
(hi )i∈I · (hi0 )i∈I = (hi hi0 )i∈I .
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We shall generalize the notion of a direct product to arbitrary families of groups in the following way:
Definition If {Hi }i∈I is a family Q of groups, its direct product, denoted ×i∈I Hi is the cartesian product Hi , endorsed with the binary operation defined by i∈I
(hi )i∈I · (hi0 )i∈I = (hi hi0 )i∈I .
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The following characterisation of finite direct products is often useful:
Proposition Let (G , ·) be a group and H1 , H2 , . . . , Hn E G . Then G = H1 × H2 × . . . × Hn if and only if G = H 1 H2 . . . Hn , (H1 H2 . . . Hi−1 ) ∩ Hi = 1, (∀)i = 2, n .
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The following characterisation of finite direct products is often useful:
Proposition Let (G , ·) be a group and H1 , H2 , . . . , Hn E G . Then G = H1 × H2 × . . . × Hn if and only if G = H 1 H2 . . . Hn , (H1 H2 . . . Hi−1 ) ∩ Hi = 1, (∀)i = 2, n .
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Semidirect products Definition Let (G , ·) be a group and K E G . A subgroup H ≤ G is called a complement of the subgroup K in G if G = HK and H ∩ K = 1.
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Semidirect products Definition Let (G , ·) be a group and K E G . A subgroup H ≤ G is called a complement of the subgroup K in G if G = HK and H ∩ K = 1.
Remark 1) If the complement H is also normal in G , H is called a direct complement of K in G and in this case G = H × K is the direct product of H and K .
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Semidirect products Definition Let (G , ·) be a group and K E G . A subgroup H ≤ G is called a complement of the subgroup K in G if G = HK and H ∩ K = 1.
Remark 1) If the complement H is also normal in G , H is called a direct complement of K in G and in this case G = H × K is the direct product of H and K . 2) If H is a complement of the normal subgroup K in the group G , then any element g ∈ G can be uniquely written as g = hk, with h ∈ H and k ∈ K .
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Semidirect products Definition Let (G , ·) be a group and K E G . A subgroup H ≤ G is called a complement of the subgroup K in G if G = HK and H ∩ K = 1.
Remark 1) If the complement H is also normal in G , H is called a direct complement of K in G and in this case G = H × K is the direct product of H and K . 2) If H is a complement of the normal subgroup K in the group G , then any element g ∈ G can be uniquely written as g = hk, with h ∈ H and k ∈ K . 3) If H is a complement of the normal subgroup K in the group G , and g1 = h1 k1 , g2 = h2 k2 ∈ G , we have g1 g2 = (h1 k1 )(h2 k2 ) = h1 h2 · (h2−1 k1 h2 )k2 .
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Semidirect products Definition Let (G , ·) be a group and K E G . A subgroup H ≤ G is called a complement of the subgroup K in G if G = HK and H ∩ K = 1.
Remark 1) If the complement H is also normal in G , H is called a direct complement of K in G and in this case G = H × K is the direct product of H and K . 2) If H is a complement of the normal subgroup K in the group G , then any element g ∈ G can be uniquely written as g = hk, with h ∈ H and k ∈ K . 3) If H is a complement of the normal subgroup K in the group G , and g1 = h1 k1 , g2 = h2 k2 ∈ G , we have g1 g2 = (h1 k1 )(h2 k2 ) = h1 h2 · (h2−1 k1 h2 )k2 .
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Proposition Let (H, ·) and (K , ·) be two groups and ϕ : H −→ Aut(K ) a group homomorphism. Then the binary operation defined on the cartesian product H × K by ϕ
(h1 , k1 ) · (h2 , k2 ) = (h1 h2 , (k1 )(h2 ) k2 ) determines on H × K a group structure.
Definition The group defined in the previous proposition is called the semidirect product of the group H with the group K via the homomorphism ϕ and is denoted H nϕ K .
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Proposition Let (H, ·) and (K , ·) be two groups and ϕ : H −→ Aut(K ) a group homomorphism. Then the binary operation defined on the cartesian product H × K by ϕ
(h1 , k1 ) · (h2 , k2 ) = (h1 h2 , (k1 )(h2 ) k2 ) determines on H × K a group structure.
Definition The group defined in the previous proposition is called the semidirect product of the group H with the group K via the homomorphism ϕ and is denoted H nϕ K .
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Remark Similar to the case of direct products, considering the semidirect product b = H × {1K } and K b represent two subgroups H nϕ K , the subsets H b K b ≤ H nϕ K , with the properties: H, b∼ H = H,
b ∼ K = K,
M.Chi¸s ()
b E H nϕ K , H
Lecture 5
bK b, H nϕ K = H
b ∩K b = 1. H
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Group actions
Definition Let (G , ·) be a group and M 6= ∅ a nonempty set. An application α : G × M −→ M is called a (left) action of G on M if it satisfies the properties: 1) (gh, m)α = (g , (h, m)α )α , (∀)g , h ∈ G , m ∈ M . 2) (1, m)α = m , (∀)m ∈ M .
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Group actions
Definition Let (G , ·) be a group and M 6= ∅ a nonempty set. An application α : G × M −→ M is called a (left) action of G on M if it satisfies the properties: 1) (gh, m)α = (g , (h, m)α )α , (∀)g , h ∈ G , m ∈ M . 2) (1, m)α = m , (∀)m ∈ M .
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Remark 1) If we denote g · m := (g , m)α , the above conditions can be rewritten as: 1) (gh) · m = g · (h · m) , (∀)g , h ∈ G , m ∈ M . 2) 1 · m = m , (∀)m ∈ M . 2) In a similar way one can define the notion of a right action of a group on a nonempty set, for which one can prove similar properties to those we shall prove about left actions. In the sequel we shall only refer to left actions, which we shall call simply actions.
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Remark 1) If we denote g · m := (g , m)α , the above conditions can be rewritten as: 1) (gh) · m = g · (h · m) , (∀)g , h ∈ G , m ∈ M . 2) 1 · m = m , (∀)m ∈ M . 2) In a similar way one can define the notion of a right action of a group on a nonempty set, for which one can prove similar properties to those we shall prove about left actions. In the sequel we shall only refer to left actions, which we shall call simply actions.
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Definition not
If α : G × M −→ M : (g , m) 7−→ (g , m)α = g · m is an action, we define on M the association relation ∼α with respect to α by def
x ∼α y ⇐⇒ (∃)g ∈ G : g · x = y .
Proposition not
If α : G × M −→ M : (g , m) 7−→ (g , m)α = g · m is an action, the association relation with respect to the action α is an equivalence relation on M.
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Definition not
If α : G × M −→ M : (g , m) 7−→ (g , m)α = g · m is an action, we define on M the association relation ∼α with respect to α by def
x ∼α y ⇐⇒ (∃)g ∈ G : g · x = y .
Proposition not
If α : G × M −→ M : (g , m) 7−→ (g , m)α = g · m is an action, the association relation with respect to the action α is an equivalence relation on M.
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Definition The equivalence class of an element x ∈ M with respect to the relation ∼α is called the orbit of the element x with respect to the action α.
Remark The orbit of an element x ∈ M withe respect to an action α : G × M −→ M is [x]∼α = {y ∈ M| x ∼α y } = {y ∈ M| (∃)g ∈ G : y = g · x} = not = {g · x| g ∈ G } = G · x .
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Definition The equivalence class of an element x ∈ M with respect to the relation ∼α is called the orbit of the element x with respect to the action α.
Remark The orbit of an element x ∈ M withe respect to an action α : G × M −→ M is [x]∼α = {y ∈ M| x ∼α y } = {y ∈ M| (∃)g ∈ G : y = g · x} = not = {g · x| g ∈ G } = G · x .
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Definition The stabilizer of an element x ∈ M with respect to an action α : G × M −→ M is the set StabG (x) := {g ∈ G | g · x = x} .
Proposition The stabilizer StabG (x) of an element x ∈ M with respect to an action α : G × M −→ M of a group (G , ·) on a set M is a subgroup of the group G.
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Definition The stabilizer of an element x ∈ M with respect to an action α : G × M −→ M is the set StabG (x) := {g ∈ G | g · x = x} .
Proposition The stabilizer StabG (x) of an element x ∈ M with respect to an action α : G × M −→ M of a group (G , ·) on a set M is a subgroup of the group G.
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Proposition The cardinal of the orbit of an element x ∈ M with respect to an action α : G × M −→ M is equal to the index of the stabilizer of the element: |G · x| = [G : StabG (x)] .
Corollary If M is a finite set, and R a representative system of the orbits defined by the action α : G × M −→ M on M, then X |M| = [G : StabG (x)] x∈R
(the class equation associated to the action α).
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Proposition The cardinal of the orbit of an element x ∈ M with respect to an action α : G × M −→ M is equal to the index of the stabilizer of the element: |G · x| = [G : StabG (x)] .
Corollary If M is a finite set, and R a representative system of the orbits defined by the action α : G × M −→ M on M, then X |M| = [G : StabG (x)] x∈R
(the class equation associated to the action α).
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Definition If α : G × M −→ M is an action a of a group G on a set M, and g ∈ G , we denote by Fix(g ) the set of fixed points with respect to g by the action α: Fix(g ) := {x ∈ M| g · x = x} .
Remark For x ∈ M and g ∈ G we have x ∈ Fix(g ) ⇐⇒ g · x = x ⇐⇒ g ∈ StabG (x) .
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Definition If α : G × M −→ M is an action a of a group G on a set M, and g ∈ G , we denote by Fix(g ) the set of fixed points with respect to g by the action α: Fix(g ) := {x ∈ M| g · x = x} .
Remark For x ∈ M and g ∈ G we have x ∈ Fix(g ) ⇐⇒ g · x = x ⇐⇒ g ∈ StabG (x) .
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Proposition Let α : G × M −→ M be an action of a finite group G on a finite set M. Then number n of orbits determined by the action α in M is n=
1 X Fix(g )| |G | g ∈G
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Two structure theorems Sylow’s theorem
M.Chi¸s ()
Lecture 5
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Two structure theorems Sylow’s theorem
Definition Let p be a prime. A finite group (G , ·) is called p−group if its ordiner is an integer power of the prime number p: (∃)k ∈ N :
M.Chi¸s ()
|G | = p k .
Lecture 5
3.XI.2008
23 / 26
Two structure theorems Sylow’s theorem
Definition Let p be a prime. A finite group (G , ·) is called p−group if its ordiner is an integer power of the prime number p: (∃)k ∈ N :
|G | = p k .
If we denote by π(G ) the set of prime divisors of the order |G | of the group G , π(G ) = {p ∈ N| p − prime, p| |G |} then G is a p−group if and only if π(G ) = {p }.
M.Chi¸s ()
Lecture 5
3.XI.2008
23 / 26
Two structure theorems Sylow’s theorem
Definition Let p be a prime. A finite group (G , ·) is called p−group if its ordiner is an integer power of the prime number p: (∃)k ∈ N :
|G | = p k .
If we denote by π(G ) the set of prime divisors of the order |G | of the group G , π(G ) = {p ∈ N| p − prime, p| |G |} then G is a p−group if and only if π(G ) = {p }. Generally, if π is a set of prime numbers, a finite group (G , ·) is called a π−group if π(G ) ⊆ π. M.Chi¸s ()
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Two structure theorems Sylow’s theorem
Definition Let p be a prime. A finite group (G , ·) is called p−group if its ordiner is an integer power of the prime number p: (∃)k ∈ N :
|G | = p k .
If we denote by π(G ) the set of prime divisors of the order |G | of the group G , π(G ) = {p ∈ N| p − prime, p| |G |} then G is a p−group if and only if π(G ) = {p }. Generally, if π is a set of prime numbers, a finite group (G , ·) is called a π−group if π(G ) ⊆ π. M.Chi¸s ()
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Definition Let p be a prime. A finite subgroup H ≤ G of a group G is called a p−subgroup of the group G if H is a p−group. Generally, if π is a set of prime numbers, H is called a π−subgroup of the group G if H is a π−group.
Definition If the group G is finite and |G | = p k m, with k, m ∈ N, m 6= 0, p 6 |m, a p−subgroup H of G is called a p−Sylow subgroup of G if |H| = p k . We denote by Sylp (G ) the set of p−Sylow subgroups of a group G . Also, we denote k = vp (G ), so that Sylp (G ) = {H ≤ G | |H| = p vp (G ) } .
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Definition Let p be a prime. A finite subgroup H ≤ G of a group G is called a p−subgroup of the group G if H is a p−group. Generally, if π is a set of prime numbers, H is called a π−subgroup of the group G if H is a π−group.
Definition If the group G is finite and |G | = p k m, with k, m ∈ N, m 6= 0, p 6 |m, a p−subgroup H of G is called a p−Sylow subgroup of G if |H| = p k . We denote by Sylp (G ) the set of p−Sylow subgroups of a group G . Also, we denote k = vp (G ), so that Sylp (G ) = {H ≤ G | |H| = p vp (G ) } .
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Proposition Sylow’s theorem Let (G , ·) be a finite group, and p a prime number. Then the following properties(traditionally called Sylow’s theorems) hold: 1) If for some k ∈ N, p k | |G |, then the number N(p k ) of subgroups of order p k of G satisfies the congruence N(p k ) ≡ 1 (mod p) .
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Proposition Sylow’s theorem Let (G , ·) be a finite group, and p a prime number. Then the following properties(traditionally called Sylow’s theorems) hold: 1) If for some k ∈ N, p k | |G |, then the number N(p k ) of subgroups of order p k of G satisfies the congruence N(p k ) ≡ 1 (mod p) . In particular, there are p−subgroups of any order p k , dividing the order |G | of the group G . In particular, Sylp (G ) 6= ∅.
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Proposition Sylow’s theorem Let (G , ·) be a finite group, and p a prime number. Then the following properties(traditionally called Sylow’s theorems) hold: 1) If for some k ∈ N, p k | |G |, then the number N(p k ) of subgroups of order p k of G satisfies the congruence N(p k ) ≡ 1 (mod p) . In particular, there are p−subgroups of any order p k , dividing the order |G | of the group G . In particular, Sylp (G ) 6= ∅. 2) If P ∈ Sylp (G ), and H is a p−subgroup of G , then there is g ∈ G such that H ⊆ P g (:= g −1 Pg ). In particular, all p−Sylow subgroups of the group G are conjugated with each other.
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Proposition Sylow’s theorem Let (G , ·) be a finite group, and p a prime number. Then the following properties(traditionally called Sylow’s theorems) hold: 1) If for some k ∈ N, p k | |G |, then the number N(p k ) of subgroups of order p k of G satisfies the congruence N(p k ) ≡ 1 (mod p) . In particular, there are p−subgroups of any order p k , dividing the order |G | of the group G . In particular, Sylp (G ) 6= ∅. 2) If P ∈ Sylp (G ), and H is a p−subgroup of G , then there is g ∈ G such that H ⊆ P g (:= g −1 Pg ). In particular, all p−Sylow subgroups of the group G are conjugated with each other. 3) If P ∈ Sylp (G ), and np = |Sylp (G )|, then np = [G : NG (P)] and np ≡ 1(mod p).
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Proposition Sylow’s theorem Let (G , ·) be a finite group, and p a prime number. Then the following properties(traditionally called Sylow’s theorems) hold: 1) If for some k ∈ N, p k | |G |, then the number N(p k ) of subgroups of order p k of G satisfies the congruence N(p k ) ≡ 1 (mod p) . In particular, there are p−subgroups of any order p k , dividing the order |G | of the group G . In particular, Sylp (G ) 6= ∅. 2) If P ∈ Sylp (G ), and H is a p−subgroup of G , then there is g ∈ G such that H ⊆ P g (:= g −1 Pg ). In particular, all p−Sylow subgroups of the group G are conjugated with each other. 3) If P ∈ Sylp (G ), and np = |Sylp (G )|, then np = [G : NG (P)] and np ≡ 1(mod p).
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25 / 26
The structure theorem for finitely generated abelian groups
Proposition Let (G , ·) be a finitely generated abelian groups. Then there are nonnegative integers m, n ∈ N, m ≤ n, and d1 , d2 , . . . , dm ∈ N with d1 ≥ 2 and di |di+1 , (∀)i = 1, m − 1, such that G∼ = Zd1 × Zd2 × . . . × Zdm × Zn−m . (n represents the minimal number of generators of the abelian group G ).
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Lecture 5
3.XI.2008
26 / 26
The structure theorem for finitely generated abelian groups
Proposition Let (G , ·) be a finitely generated abelian groups. Then there are nonnegative integers m, n ∈ N, m ≤ n, and d1 , d2 , . . . , dm ∈ N with d1 ≥ 2 and di |di+1 , (∀)i = 1, m − 1, such that G∼ = Zd1 × Zd2 × . . . × Zdm × Zn−m . (n represents the minimal number of generators of the abelian group G ). In particular, if G is a finite abelian group, then there are nonnegative integers m ∈ N and d1 , d2 , . . . , dm ∈ N with d1 ≥ 2 and di |di+1 , (∀)i = 1, m − 1, such that G∼ = Zd1 × Zd2 × . . . × Zdm .
M.Chi¸s ()
Lecture 5
3.XI.2008
26 / 26
The structure theorem for finitely generated abelian groups
Proposition Let (G , ·) be a finitely generated abelian groups. Then there are nonnegative integers m, n ∈ N, m ≤ n, and d1 , d2 , . . . , dm ∈ N with d1 ≥ 2 and di |di+1 , (∀)i = 1, m − 1, such that G∼ = Zd1 × Zd2 × . . . × Zdm × Zn−m . (n represents the minimal number of generators of the abelian group G ). In particular, if G is a finite abelian group, then there are nonnegative integers m ∈ N and d1 , d2 , . . . , dm ∈ N with d1 ≥ 2 and di |di+1 , (∀)i = 1, m − 1, such that G∼ = Zd1 × Zd2 × . . . × Zdm .
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The isomorphism theorems
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Groups(sequel)
The isomorphism theorems Proposition (The fundamental isomorphism theorem) If f : (G , ·) −→ (T , ·) is a group homomorphism, then there is an unique isomorphism f : G /Ker (f ) −→ Im(f ) , such that πKer (f ) · f · iIm(f ) = f .
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Groups(sequel)
The isomorphism theorems Proposition (The fundamental isomorphism theorem) If f : (G , ·) −→ (T , ·) is a group homomorphism, then there is an unique isomorphism f : G /Ker (f ) −→ Im(f ) , such that πKer (f ) · f · iIm(f ) = f .
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Remark The isomorphism f : G /Ker (f ) −→ Im(f ) is the unique application which makes the following diagram commutative: G π↓
f
−−−−→
T ↑i
f
G /Ker (f ) −−−−→ Im(f )
Corollary If f : (G , ·) −→ (T , ·) is a group homomorphism, then G /Ker (f ) ∼ = Im(f ) .
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Remark The isomorphism f : G /Ker (f ) −→ Im(f ) is the unique application which makes the following diagram commutative: G π↓
f
−−−−→
T ↑i
f
G /Ker (f ) −−−−→ Im(f )
Corollary If f : (G , ·) −→ (T , ·) is a group homomorphism, then G /Ker (f ) ∼ = Im(f ) .
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Proposition (The first isomorphism theorem) If f : (G , ·) −→ (T , ·) is a surjective group homomorphism, H E G with Ker (f ) ⊆ H, and U = (H)f , then G /H ∼ = T /U.
Proposition (The second isomorphism theorem) Let (G , ·) be a group, H ≤ G and N E G . Then H ∩ N E H and H/(H ∩ N) ∼ = HN/N .
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Proposition (The first isomorphism theorem) If f : (G , ·) −→ (T , ·) is a surjective group homomorphism, H E G with Ker (f ) ⊆ H, and U = (H)f , then G /H ∼ = T /U.
Proposition (The second isomorphism theorem) Let (G , ·) be a group, H ≤ G and N E G . Then H ∩ N E H and H/(H ∩ N) ∼ = HN/N .
Proposition (The third isomorphism theorem) Let (G , ·) be a group and K , N E G , with K ⊆ N. Then N/K E G /K and (G /K )/(N/K ) ∼ = G /N . M.Chi¸s ()
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Proposition (The first isomorphism theorem) If f : (G , ·) −→ (T , ·) is a surjective group homomorphism, H E G with Ker (f ) ⊆ H, and U = (H)f , then G /H ∼ = T /U.
Proposition (The second isomorphism theorem) Let (G , ·) be a group, H ≤ G and N E G . Then H ∩ N E H and H/(H ∩ N) ∼ = HN/N .
Proposition (The third isomorphism theorem) Let (G , ·) be a group and K , N E G , with K ⊆ N. Then N/K E G /K and (G /K )/(N/K ) ∼ = G /N . M.Chi¸s ()
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Proposition (The corespondence theorem for groups) Let (G , ·) be a group, K E G and πK : G −→ G /K the canonical projection. The function πK establishes then a bijective corespondence M ←→ M ∗ between the set of subgroups of G which include K and the set of all subgroups of G /K . Also, for any L, M ≤ G with K ⊆ L, M we have
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Proposition (The corespondence theorem for groups) Let (G , ·) be a group, K E G and πK : G −→ G /K the canonical projection. The function πK establishes then a bijective corespondence M ←→ M ∗ between the set of subgroups of G which include K and the set of all subgroups of G /K . Also, for any L, M ≤ G with K ⊆ L, M we have 1) M ∗ = M/K = (M)πK .
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Proposition (The corespondence theorem for groups) Let (G , ·) be a group, K E G and πK : G −→ G /K the canonical projection. The function πK establishes then a bijective corespondence M ←→ M ∗ between the set of subgroups of G which include K and the set of all subgroups of G /K . Also, for any L, M ≤ G with K ⊆ L, M we have 1) M ∗ = M/K = (M)πK . 2) L ⊆ M ⇐⇒ L∗ ⊆ M ∗ , and in this case [M : L] = [M ∗ : L∗ ].
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Proposition (The corespondence theorem for groups) Let (G , ·) be a group, K E G and πK : G −→ G /K the canonical projection. The function πK establishes then a bijective corespondence M ←→ M ∗ between the set of subgroups of G which include K and the set of all subgroups of G /K . Also, for any L, M ≤ G with K ⊆ L, M we have 1) M ∗ = M/K = (M)πK . 2) L ⊆ M ⇐⇒ L∗ ⊆ M ∗ , and in this case [M : L] = [M ∗ : L∗ ]. 3) L E M ⇐⇒ L∗ E M ∗ , and in this case M/L ∼ = M ∗ /L∗ .
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Proposition (The corespondence theorem for groups) Let (G , ·) be a group, K E G and πK : G −→ G /K the canonical projection. The function πK establishes then a bijective corespondence M ←→ M ∗ between the set of subgroups of G which include K and the set of all subgroups of G /K . Also, for any L, M ≤ G with K ⊆ L, M we have 1) M ∗ = M/K = (M)πK . 2) L ⊆ M ⇐⇒ L∗ ⊆ M ∗ , and in this case [M : L] = [M ∗ : L∗ ]. 3) L E M ⇐⇒ L∗ E M ∗ , and in this case M/L ∼ = M ∗ /L∗ .
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3.XI.2008
4 / 26
Cyclic groups
Proposition Let (Z, +) be the additive group of the integers, and H ≤ Z. Then there is a nonnegative integer n ∈ N such that H = nZ.
Remark Let n ∈ N∗ be a positive integer. Then the binary relations on Z of congruence modulo n, respectively modulo the subgroup nZ, coincide: k ≡ l(mod n) ⇐⇒ n|k − l ⇐⇒ k − l ∈ nZ ⇐⇒ k ≡ l(mod nZ) . Hence, Zn := Z/≡(mod n) = Z/nZ.
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Lecture 5
3.XI.2008
5 / 26
Cyclic groups
Proposition Let (Z, +) be the additive group of the integers, and H ≤ Z. Then there is a nonnegative integer n ∈ N such that H = nZ.
Remark Let n ∈ N∗ be a positive integer. Then the binary relations on Z of congruence modulo n, respectively modulo the subgroup nZ, coincide: k ≡ l(mod n) ⇐⇒ n|k − l ⇐⇒ k − l ∈ nZ ⇐⇒ k ≡ l(mod nZ) . Hence, Zn := Z/≡(mod n) = Z/nZ.
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Proposition Let (G , ·) be a cyclic group. Then either G ∼ = Z, or there is a positive integer n ∈ N∗ such that G ∼ Z . = n
Proposition Let (G , ·) be a finite cyclic group, of order |G | = n.
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Proposition Let (G , ·) be a cyclic group. Then either G ∼ = Z, or there is a positive integer n ∈ N∗ such that G ∼ Z . = n
Proposition Let (G , ·) be a finite cyclic group, of order |G | = n. 1) If H ≤ G , then there is a positive integer d ∈ N∗ , with d|n, such that H∼ = db · Zn .
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Proposition Let (G , ·) be a cyclic group. Then either G ∼ = Z, or there is a positive integer n ∈ N∗ such that G ∼ Z . = n
Proposition Let (G , ·) be a finite cyclic group, of order |G | = n. 1) If H ≤ G , then there is a positive integer d ∈ N∗ , with d|n, such that H∼ = db · Zn . 2) For any divisor d ∈ N∗ of the order n of the group, G has a unique subgroup of order d.
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Proposition Let (G , ·) be a cyclic group. Then either G ∼ = Z, or there is a positive integer n ∈ N∗ such that G ∼ Z . = n
Proposition Let (G , ·) be a finite cyclic group, of order |G | = n. 1) If H ≤ G , then there is a positive integer d ∈ N∗ , with d|n, such that H∼ = db · Zn . 2) For any divisor d ∈ N∗ of the order n of the group, G has a unique subgroup of order d. 3) The number of elements a ∈ G such that G = hai is (n)ϕ , where ϕ is Euler’s function(given by (n)ϕ =the number of nonnegative integers k, with 0 ≤ k ≤ n − 1, such that (k, n) = 1).
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6 / 26
Proposition Let (G , ·) be a cyclic group. Then either G ∼ = Z, or there is a positive integer n ∈ N∗ such that G ∼ Z . = n
Proposition Let (G , ·) be a finite cyclic group, of order |G | = n. 1) If H ≤ G , then there is a positive integer d ∈ N∗ , with d|n, such that H∼ = db · Zn . 2) For any divisor d ∈ N∗ of the order n of the group, G has a unique subgroup of order d. 3) The number of elements a ∈ G such that G = hai is (n)ϕ , where ϕ is Euler’s function(given by (n)ϕ =the number of nonnegative integers k, with P0 ≤ k ≤ n − 1, such that (k, n) = 1). 4) (d)ϕ = n. d|n
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6 / 26
Proposition Let (G , ·) be a cyclic group. Then either G ∼ = Z, or there is a positive integer n ∈ N∗ such that G ∼ Z . = n
Proposition Let (G , ·) be a finite cyclic group, of order |G | = n. 1) If H ≤ G , then there is a positive integer d ∈ N∗ , with d|n, such that H∼ = db · Zn . 2) For any divisor d ∈ N∗ of the order n of the group, G has a unique subgroup of order d. 3) The number of elements a ∈ G such that G = hai is (n)ϕ , where ϕ is Euler’s function(given by (n)ϕ =the number of nonnegative integers k, with P0 ≤ k ≤ n − 1, such that (k, n) = 1). 4) (d)ϕ = n. d|n
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6 / 26
Direct and semidirect products of groups
Direct products
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Direct and semidirect products of groups
Direct products Definition Let (G , ·) be a group and H, K E G . G is called the internal direct product of its normal subgroups H and K , if G = HK and H ∩ K = 1.
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7 / 26
Direct and semidirect products of groups
Direct products Definition Let (G , ·) be a group and H, K E G . G is called the internal direct product of its normal subgroups H and K , if G = HK and H ∩ K = 1.
Proposition If (G , ·) is the direct product of its normal subgroups H and K , then hk = kh, (∀)h ∈ H, k ∈ K and for any g ∈ G there are unique elements h ∈ H and k ∈ K such that g = hk.
M.Chi¸s ()
Lecture 5
3.XI.2008
7 / 26
Direct and semidirect products of groups
Direct products Definition Let (G , ·) be a group and H, K E G . G is called the internal direct product of its normal subgroups H and K , if G = HK and H ∩ K = 1.
Proposition If (G , ·) is the direct product of its normal subgroups H and K , then hk = kh, (∀)h ∈ H, k ∈ K and for any g ∈ G there are unique elements h ∈ H and k ∈ K such that g = hk.
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Remark If (G , ·) is the direct product of its normal subgroups H and K , and g1 = h1 k1 , g2 = h2 = k2 , with h1 , h2 ∈ H, k1 , k2 ∈ K , then g1 g2 = (h1 k1 )(h2 k2 ) = h1 (k1 h1 )k2 = h1 (h2 k1 )k2 = (h1 h2 )(k1 k2 ) .
Proposition Let (H, ·) and (K , ·) be two groups. Then the binary operation defined on the cartesian product H × K by (h1 , k1 ) · (h2 , k2 ) = (h1 h2 , k1 k2 ) defines on H × K a structure of a group.
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Remark If (G , ·) is the direct product of its normal subgroups H and K , and g1 = h1 k1 , g2 = h2 = k2 , with h1 , h2 ∈ H, k1 , k2 ∈ K , then g1 g2 = (h1 k1 )(h2 k2 ) = h1 (k1 h1 )k2 = h1 (h2 k1 )k2 = (h1 h2 )(k1 k2 ) .
Proposition Let (H, ·) and (K , ·) be two groups. Then the binary operation defined on the cartesian product H × K by (h1 , k1 ) · (h2 , k2 ) = (h1 h2 , k1 k2 ) defines on H × K a structure of a group.
Definition Let (H, ·) and (K , ·) be two groups. The group defined in the previous proposition is called the external direct product of the groups H and K . M.Chi¸s ()
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Remark If (G , ·) is the direct product of its normal subgroups H and K , and g1 = h1 k1 , g2 = h2 = k2 , with h1 , h2 ∈ H, k1 , k2 ∈ K , then g1 g2 = (h1 k1 )(h2 k2 ) = h1 (k1 h1 )k2 = h1 (h2 k1 )k2 = (h1 h2 )(k1 k2 ) .
Proposition Let (H, ·) and (K , ·) be two groups. Then the binary operation defined on the cartesian product H × K by (h1 , k1 ) · (h2 , k2 ) = (h1 h2 , k1 k2 ) defines on H × K a structure of a group.
Definition Let (H, ·) and (K , ·) be two groups. The group defined in the previous proposition is called the external direct product of the groups H and K . M.Chi¸s ()
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Proposition Let (H × K , ·) be the external direct product of the groups (H, ·) and b K b ⊆ H × K the subsets of the cartesian product H × K (K , ·), and H, defined by b := H × {1K } = {(h, 1K )| h ∈ H} , H b := {1H } × K = {(1H , k)| k ∈ K } . K b K b E H × K, H ∼ b K∼ b, H × K = H bK b and H b ∩K b = 1. Then H, = H, =K
Remark According to the previous proposition, the external direct product b and K b H × K is the internal direct product of the isomorphic copies H of the groups H and K . Consequently, we can identify the notions of internal and external direct product. If a group (G , ·) is the internal direct product of its subgroups H, K E G , we write G = H × K and we shall simply call G the direct product of H and K .
M.Chi¸s ()
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3.XI.2008
9 / 26
Proposition Let (H × K , ·) be the external direct product of the groups (H, ·) and b K b ⊆ H × K the subsets of the cartesian product H × K (K , ·), and H, defined by b := H × {1K } = {(h, 1K )| h ∈ H} , H b := {1H } × K = {(1H , k)| k ∈ K } . K b K b E H × K, H ∼ b K∼ b, H × K = H bK b and H b ∩K b = 1. Then H, = H, =K
Remark According to the previous proposition, the external direct product b and K b H × K is the internal direct product of the isomorphic copies H of the groups H and K . Consequently, we can identify the notions of internal and external direct product. If a group (G , ·) is the internal direct product of its subgroups H, K E G , we write G = H × K and we shall simply call G the direct product of H and K .
M.Chi¸s ()
Lecture 5
3.XI.2008
9 / 26
We shall generalize the notion of a direct product to arbitrary families of groups in the following way:
Definition If {Hi }i∈I is a family Q of groups, its direct product, denoted ×i∈I Hi is the cartesian product Hi , endorsed with the binary operation defined by i∈I
(hi )i∈I · (hi0 )i∈I = (hi hi0 )i∈I .
M.Chi¸s ()
Lecture 5
3.XI.2008
10 / 26
We shall generalize the notion of a direct product to arbitrary families of groups in the following way:
Definition If {Hi }i∈I is a family Q of groups, its direct product, denoted ×i∈I Hi is the cartesian product Hi , endorsed with the binary operation defined by i∈I
(hi )i∈I · (hi0 )i∈I = (hi hi0 )i∈I .
M.Chi¸s ()
Lecture 5
3.XI.2008
10 / 26
The following characterisation of finite direct products is often useful:
Proposition Let (G , ·) be a group and H1 , H2 , . . . , Hn E G . Then G = H1 × H2 × . . . × Hn if and only if G = H 1 H2 . . . Hn , (H1 H2 . . . Hi−1 ) ∩ Hi = 1, (∀)i = 2, n .
M.Chi¸s ()
Lecture 5
3.XI.2008
11 / 26
The following characterisation of finite direct products is often useful:
Proposition Let (G , ·) be a group and H1 , H2 , . . . , Hn E G . Then G = H1 × H2 × . . . × Hn if and only if G = H 1 H2 . . . Hn , (H1 H2 . . . Hi−1 ) ∩ Hi = 1, (∀)i = 2, n .
M.Chi¸s ()
Lecture 5
3.XI.2008
11 / 26
Semidirect products Definition Let (G , ·) be a group and K E G . A subgroup H ≤ G is called a complement of the subgroup K in G if G = HK and H ∩ K = 1.
M.Chi¸s ()
Lecture 5
3.XI.2008
12 / 26
Semidirect products Definition Let (G , ·) be a group and K E G . A subgroup H ≤ G is called a complement of the subgroup K in G if G = HK and H ∩ K = 1.
Remark 1) If the complement H is also normal in G , H is called a direct complement of K in G and in this case G = H × K is the direct product of H and K .
M.Chi¸s ()
Lecture 5
3.XI.2008
12 / 26
Semidirect products Definition Let (G , ·) be a group and K E G . A subgroup H ≤ G is called a complement of the subgroup K in G if G = HK and H ∩ K = 1.
Remark 1) If the complement H is also normal in G , H is called a direct complement of K in G and in this case G = H × K is the direct product of H and K . 2) If H is a complement of the normal subgroup K in the group G , then any element g ∈ G can be uniquely written as g = hk, with h ∈ H and k ∈ K .
M.Chi¸s ()
Lecture 5
3.XI.2008
12 / 26
Semidirect products Definition Let (G , ·) be a group and K E G . A subgroup H ≤ G is called a complement of the subgroup K in G if G = HK and H ∩ K = 1.
Remark 1) If the complement H is also normal in G , H is called a direct complement of K in G and in this case G = H × K is the direct product of H and K . 2) If H is a complement of the normal subgroup K in the group G , then any element g ∈ G can be uniquely written as g = hk, with h ∈ H and k ∈ K . 3) If H is a complement of the normal subgroup K in the group G , and g1 = h1 k1 , g2 = h2 k2 ∈ G , we have g1 g2 = (h1 k1 )(h2 k2 ) = h1 h2 · (h2−1 k1 h2 )k2 .
M.Chi¸s ()
Lecture 5
3.XI.2008
12 / 26
Semidirect products Definition Let (G , ·) be a group and K E G . A subgroup H ≤ G is called a complement of the subgroup K in G if G = HK and H ∩ K = 1.
Remark 1) If the complement H is also normal in G , H is called a direct complement of K in G and in this case G = H × K is the direct product of H and K . 2) If H is a complement of the normal subgroup K in the group G , then any element g ∈ G can be uniquely written as g = hk, with h ∈ H and k ∈ K . 3) If H is a complement of the normal subgroup K in the group G , and g1 = h1 k1 , g2 = h2 k2 ∈ G , we have g1 g2 = (h1 k1 )(h2 k2 ) = h1 h2 · (h2−1 k1 h2 )k2 .
M.Chi¸s ()
Lecture 5
3.XI.2008
12 / 26
Proposition Let (H, ·) and (K , ·) be two groups and ϕ : H −→ Aut(K ) a group homomorphism. Then the binary operation defined on the cartesian product H × K by ϕ
(h1 , k1 ) · (h2 , k2 ) = (h1 h2 , (k1 )(h2 ) k2 ) determines on H × K a group structure.
Definition The group defined in the previous proposition is called the semidirect product of the group H with the group K via the homomorphism ϕ and is denoted H nϕ K .
M.Chi¸s ()
Lecture 5
3.XI.2008
13 / 26
Proposition Let (H, ·) and (K , ·) be two groups and ϕ : H −→ Aut(K ) a group homomorphism. Then the binary operation defined on the cartesian product H × K by ϕ
(h1 , k1 ) · (h2 , k2 ) = (h1 h2 , (k1 )(h2 ) k2 ) determines on H × K a group structure.
Definition The group defined in the previous proposition is called the semidirect product of the group H with the group K via the homomorphism ϕ and is denoted H nϕ K .
M.Chi¸s ()
Lecture 5
3.XI.2008
13 / 26
Remark Similar to the case of direct products, considering the semidirect product b = H × {1K } and K b represent two subgroups H nϕ K , the subsets H b K b ≤ H nϕ K , with the properties: H, b∼ H = H,
b ∼ K = K,
M.Chi¸s ()
b E H nϕ K , H
Lecture 5
bK b, H nϕ K = H
b ∩K b = 1. H
3.XI.2008
14 / 26
Group actions
Definition Let (G , ·) be a group and M 6= ∅ a nonempty set. An application α : G × M −→ M is called a (left) action of G on M if it satisfies the properties: 1) (gh, m)α = (g , (h, m)α )α , (∀)g , h ∈ G , m ∈ M . 2) (1, m)α = m , (∀)m ∈ M .
M.Chi¸s ()
Lecture 5
3.XI.2008
15 / 26
Group actions
Definition Let (G , ·) be a group and M 6= ∅ a nonempty set. An application α : G × M −→ M is called a (left) action of G on M if it satisfies the properties: 1) (gh, m)α = (g , (h, m)α )α , (∀)g , h ∈ G , m ∈ M . 2) (1, m)α = m , (∀)m ∈ M .
M.Chi¸s ()
Lecture 5
3.XI.2008
15 / 26
Remark 1) If we denote g · m := (g , m)α , the above conditions can be rewritten as: 1) (gh) · m = g · (h · m) , (∀)g , h ∈ G , m ∈ M . 2) 1 · m = m , (∀)m ∈ M . 2) In a similar way one can define the notion of a right action of a group on a nonempty set, for which one can prove similar properties to those we shall prove about left actions. In the sequel we shall only refer to left actions, which we shall call simply actions.
M.Chi¸s ()
Lecture 5
3.XI.2008
16 / 26
Remark 1) If we denote g · m := (g , m)α , the above conditions can be rewritten as: 1) (gh) · m = g · (h · m) , (∀)g , h ∈ G , m ∈ M . 2) 1 · m = m , (∀)m ∈ M . 2) In a similar way one can define the notion of a right action of a group on a nonempty set, for which one can prove similar properties to those we shall prove about left actions. In the sequel we shall only refer to left actions, which we shall call simply actions.
M.Chi¸s ()
Lecture 5
3.XI.2008
16 / 26
Definition not
If α : G × M −→ M : (g , m) 7−→ (g , m)α = g · m is an action, we define on M the association relation ∼α with respect to α by def
x ∼α y ⇐⇒ (∃)g ∈ G : g · x = y .
Proposition not
If α : G × M −→ M : (g , m) 7−→ (g , m)α = g · m is an action, the association relation with respect to the action α is an equivalence relation on M.
M.Chi¸s ()
Lecture 5
3.XI.2008
17 / 26
Definition not
If α : G × M −→ M : (g , m) 7−→ (g , m)α = g · m is an action, we define on M the association relation ∼α with respect to α by def
x ∼α y ⇐⇒ (∃)g ∈ G : g · x = y .
Proposition not
If α : G × M −→ M : (g , m) 7−→ (g , m)α = g · m is an action, the association relation with respect to the action α is an equivalence relation on M.
M.Chi¸s ()
Lecture 5
3.XI.2008
17 / 26
Definition The equivalence class of an element x ∈ M with respect to the relation ∼α is called the orbit of the element x with respect to the action α.
Remark The orbit of an element x ∈ M withe respect to an action α : G × M −→ M is [x]∼α = {y ∈ M| x ∼α y } = {y ∈ M| (∃)g ∈ G : y = g · x} = not = {g · x| g ∈ G } = G · x .
M.Chi¸s ()
Lecture 5
3.XI.2008
18 / 26
Definition The equivalence class of an element x ∈ M with respect to the relation ∼α is called the orbit of the element x with respect to the action α.
Remark The orbit of an element x ∈ M withe respect to an action α : G × M −→ M is [x]∼α = {y ∈ M| x ∼α y } = {y ∈ M| (∃)g ∈ G : y = g · x} = not = {g · x| g ∈ G } = G · x .
M.Chi¸s ()
Lecture 5
3.XI.2008
18 / 26
Definition The stabilizer of an element x ∈ M with respect to an action α : G × M −→ M is the set StabG (x) := {g ∈ G | g · x = x} .
Proposition The stabilizer StabG (x) of an element x ∈ M with respect to an action α : G × M −→ M of a group (G , ·) on a set M is a subgroup of the group G.
M.Chi¸s ()
Lecture 5
3.XI.2008
19 / 26
Definition The stabilizer of an element x ∈ M with respect to an action α : G × M −→ M is the set StabG (x) := {g ∈ G | g · x = x} .
Proposition The stabilizer StabG (x) of an element x ∈ M with respect to an action α : G × M −→ M of a group (G , ·) on a set M is a subgroup of the group G.
M.Chi¸s ()
Lecture 5
3.XI.2008
19 / 26
Proposition The cardinal of the orbit of an element x ∈ M with respect to an action α : G × M −→ M is equal to the index of the stabilizer of the element: |G · x| = [G : StabG (x)] .
Corollary If M is a finite set, and R a representative system of the orbits defined by the action α : G × M −→ M on M, then X |M| = [G : StabG (x)] x∈R
(the class equation associated to the action α).
M.Chi¸s ()
Lecture 5
3.XI.2008
20 / 26
Proposition The cardinal of the orbit of an element x ∈ M with respect to an action α : G × M −→ M is equal to the index of the stabilizer of the element: |G · x| = [G : StabG (x)] .
Corollary If M is a finite set, and R a representative system of the orbits defined by the action α : G × M −→ M on M, then X |M| = [G : StabG (x)] x∈R
(the class equation associated to the action α).
M.Chi¸s ()
Lecture 5
3.XI.2008
20 / 26
Definition If α : G × M −→ M is an action a of a group G on a set M, and g ∈ G , we denote by Fix(g ) the set of fixed points with respect to g by the action α: Fix(g ) := {x ∈ M| g · x = x} .
Remark For x ∈ M and g ∈ G we have x ∈ Fix(g ) ⇐⇒ g · x = x ⇐⇒ g ∈ StabG (x) .
M.Chi¸s ()
Lecture 5
3.XI.2008
21 / 26
Definition If α : G × M −→ M is an action a of a group G on a set M, and g ∈ G , we denote by Fix(g ) the set of fixed points with respect to g by the action α: Fix(g ) := {x ∈ M| g · x = x} .
Remark For x ∈ M and g ∈ G we have x ∈ Fix(g ) ⇐⇒ g · x = x ⇐⇒ g ∈ StabG (x) .
M.Chi¸s ()
Lecture 5
3.XI.2008
21 / 26
Proposition Let α : G × M −→ M be an action of a finite group G on a finite set M. Then number n of orbits determined by the action α in M is n=
1 X Fix(g )| |G | g ∈G
M.Chi¸s ()
Lecture 5
3.XI.2008
22 / 26
Two structure theorems Sylow’s theorem
M.Chi¸s ()
Lecture 5
3.XI.2008
23 / 26
Two structure theorems Sylow’s theorem
Definition Let p be a prime. A finite group (G , ·) is called p−group if its ordiner is an integer power of the prime number p: (∃)k ∈ N :
M.Chi¸s ()
|G | = p k .
Lecture 5
3.XI.2008
23 / 26
Two structure theorems Sylow’s theorem
Definition Let p be a prime. A finite group (G , ·) is called p−group if its ordiner is an integer power of the prime number p: (∃)k ∈ N :
|G | = p k .
If we denote by π(G ) the set of prime divisors of the order |G | of the group G , π(G ) = {p ∈ N| p − prime, p| |G |} then G is a p−group if and only if π(G ) = {p }.
M.Chi¸s ()
Lecture 5
3.XI.2008
23 / 26
Two structure theorems Sylow’s theorem
Definition Let p be a prime. A finite group (G , ·) is called p−group if its ordiner is an integer power of the prime number p: (∃)k ∈ N :
|G | = p k .
If we denote by π(G ) the set of prime divisors of the order |G | of the group G , π(G ) = {p ∈ N| p − prime, p| |G |} then G is a p−group if and only if π(G ) = {p }. Generally, if π is a set of prime numbers, a finite group (G , ·) is called a π−group if π(G ) ⊆ π. M.Chi¸s ()
Lecture 5
3.XI.2008
23 / 26
Two structure theorems Sylow’s theorem
Definition Let p be a prime. A finite group (G , ·) is called p−group if its ordiner is an integer power of the prime number p: (∃)k ∈ N :
|G | = p k .
If we denote by π(G ) the set of prime divisors of the order |G | of the group G , π(G ) = {p ∈ N| p − prime, p| |G |} then G is a p−group if and only if π(G ) = {p }. Generally, if π is a set of prime numbers, a finite group (G , ·) is called a π−group if π(G ) ⊆ π. M.Chi¸s ()
Lecture 5
3.XI.2008
23 / 26
Definition Let p be a prime. A finite subgroup H ≤ G of a group G is called a p−subgroup of the group G if H is a p−group. Generally, if π is a set of prime numbers, H is called a π−subgroup of the group G if H is a π−group.
Definition If the group G is finite and |G | = p k m, with k, m ∈ N, m 6= 0, p 6 |m, a p−subgroup H of G is called a p−Sylow subgroup of G if |H| = p k . We denote by Sylp (G ) the set of p−Sylow subgroups of a group G . Also, we denote k = vp (G ), so that Sylp (G ) = {H ≤ G | |H| = p vp (G ) } .
M.Chi¸s ()
Lecture 5
3.XI.2008
24 / 26
Definition Let p be a prime. A finite subgroup H ≤ G of a group G is called a p−subgroup of the group G if H is a p−group. Generally, if π is a set of prime numbers, H is called a π−subgroup of the group G if H is a π−group.
Definition If the group G is finite and |G | = p k m, with k, m ∈ N, m 6= 0, p 6 |m, a p−subgroup H of G is called a p−Sylow subgroup of G if |H| = p k . We denote by Sylp (G ) the set of p−Sylow subgroups of a group G . Also, we denote k = vp (G ), so that Sylp (G ) = {H ≤ G | |H| = p vp (G ) } .
M.Chi¸s ()
Lecture 5
3.XI.2008
24 / 26
Proposition Sylow’s theorem Let (G , ·) be a finite group, and p a prime number. Then the following properties(traditionally called Sylow’s theorems) hold: 1) If for some k ∈ N, p k | |G |, then the number N(p k ) of subgroups of order p k of G satisfies the congruence N(p k ) ≡ 1 (mod p) .
M.Chi¸s ()
Lecture 5
3.XI.2008
25 / 26
Proposition Sylow’s theorem Let (G , ·) be a finite group, and p a prime number. Then the following properties(traditionally called Sylow’s theorems) hold: 1) If for some k ∈ N, p k | |G |, then the number N(p k ) of subgroups of order p k of G satisfies the congruence N(p k ) ≡ 1 (mod p) . In particular, there are p−subgroups of any order p k , dividing the order |G | of the group G . In particular, Sylp (G ) 6= ∅.
M.Chi¸s ()
Lecture 5
3.XI.2008
25 / 26
Proposition Sylow’s theorem Let (G , ·) be a finite group, and p a prime number. Then the following properties(traditionally called Sylow’s theorems) hold: 1) If for some k ∈ N, p k | |G |, then the number N(p k ) of subgroups of order p k of G satisfies the congruence N(p k ) ≡ 1 (mod p) . In particular, there are p−subgroups of any order p k , dividing the order |G | of the group G . In particular, Sylp (G ) 6= ∅. 2) If P ∈ Sylp (G ), and H is a p−subgroup of G , then there is g ∈ G such that H ⊆ P g (:= g −1 Pg ). In particular, all p−Sylow subgroups of the group G are conjugated with each other.
M.Chi¸s ()
Lecture 5
3.XI.2008
25 / 26
Proposition Sylow’s theorem Let (G , ·) be a finite group, and p a prime number. Then the following properties(traditionally called Sylow’s theorems) hold: 1) If for some k ∈ N, p k | |G |, then the number N(p k ) of subgroups of order p k of G satisfies the congruence N(p k ) ≡ 1 (mod p) . In particular, there are p−subgroups of any order p k , dividing the order |G | of the group G . In particular, Sylp (G ) 6= ∅. 2) If P ∈ Sylp (G ), and H is a p−subgroup of G , then there is g ∈ G such that H ⊆ P g (:= g −1 Pg ). In particular, all p−Sylow subgroups of the group G are conjugated with each other. 3) If P ∈ Sylp (G ), and np = |Sylp (G )|, then np = [G : NG (P)] and np ≡ 1(mod p).
M.Chi¸s ()
Lecture 5
3.XI.2008
25 / 26
Proposition Sylow’s theorem Let (G , ·) be a finite group, and p a prime number. Then the following properties(traditionally called Sylow’s theorems) hold: 1) If for some k ∈ N, p k | |G |, then the number N(p k ) of subgroups of order p k of G satisfies the congruence N(p k ) ≡ 1 (mod p) . In particular, there are p−subgroups of any order p k , dividing the order |G | of the group G . In particular, Sylp (G ) 6= ∅. 2) If P ∈ Sylp (G ), and H is a p−subgroup of G , then there is g ∈ G such that H ⊆ P g (:= g −1 Pg ). In particular, all p−Sylow subgroups of the group G are conjugated with each other. 3) If P ∈ Sylp (G ), and np = |Sylp (G )|, then np = [G : NG (P)] and np ≡ 1(mod p).
M.Chi¸s ()
Lecture 5
3.XI.2008
25 / 26
The structure theorem for finitely generated abelian groups
Proposition Let (G , ·) be a finitely generated abelian groups. Then there are nonnegative integers m, n ∈ N, m ≤ n, and d1 , d2 , . . . , dm ∈ N with d1 ≥ 2 and di |di+1 , (∀)i = 1, m − 1, such that G∼ = Zd1 × Zd2 × . . . × Zdm × Zn−m . (n represents the minimal number of generators of the abelian group G ).
M.Chi¸s ()
Lecture 5
3.XI.2008
26 / 26
The structure theorem for finitely generated abelian groups
Proposition Let (G , ·) be a finitely generated abelian groups. Then there are nonnegative integers m, n ∈ N, m ≤ n, and d1 , d2 , . . . , dm ∈ N with d1 ≥ 2 and di |di+1 , (∀)i = 1, m − 1, such that G∼ = Zd1 × Zd2 × . . . × Zdm × Zn−m . (n represents the minimal number of generators of the abelian group G ). In particular, if G is a finite abelian group, then there are nonnegative integers m ∈ N and d1 , d2 , . . . , dm ∈ N with d1 ≥ 2 and di |di+1 , (∀)i = 1, m − 1, such that G∼ = Zd1 × Zd2 × . . . × Zdm .
M.Chi¸s ()
Lecture 5
3.XI.2008
26 / 26
The structure theorem for finitely generated abelian groups
Proposition Let (G , ·) be a finitely generated abelian groups. Then there are nonnegative integers m, n ∈ N, m ≤ n, and d1 , d2 , . . . , dm ∈ N with d1 ≥ 2 and di |di+1 , (∀)i = 1, m − 1, such that G∼ = Zd1 × Zd2 × . . . × Zdm × Zn−m . (n represents the minimal number of generators of the abelian group G ). In particular, if G is a finite abelian group, then there are nonnegative integers m ∈ N and d1 , d2 , . . . , dm ∈ N with d1 ≥ 2 and di |di+1 , (∀)i = 1, m − 1, such that G∼ = Zd1 × Zd2 × . . . × Zdm .
M.Chi¸s ()
Lecture 5
3.XI.2008
26 / 26
