Tema 6
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6
Temˇ a
P. 6.1. Fie H ¸si K douˇ a grupuri (multiplicative) ¸si G = H × K. Arˇatat¸i cˇa G este un grup ˆın raport cu operat¸ia binarˇ a · : G × G −→ G definitˇ a prin (h1 , k1 ) · (h2 , k2 ) = (h1 h2 , k1 k2 ) , (∀)(h1 , k1 ), (h2 , k2 ) ∈ G . (produsul direct al grupurilor H ¸si K) P. 6.2. Fie H ¸si K douˇ a grupuri (multiplicative), ϕ : H −→ Aut(K) un morfism de grupuri, iar G = H × K. Arˇ atat¸i cˇ a G este un grup ˆın raport cu operat¸ia binarˇa · : G × G −→ G definitˇa prin ϕ
(h1 , k1 ) · (h2 , k2 ) = (h1 h2 , (k1 )(h2 ) k2 ) , (∀)(h1 , k1 ), (h2 , k2 ) ∈ G . (produsul semidirect al grupurilor H ¸si K via morfismul ϕ)
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Temˇ a
P. 6.1. Fie H ¸si K douˇ a grupuri (multiplicative) ¸si G = H × K. Arˇatat¸i cˇa G este un grup ˆın raport cu operat¸ia binarˇ a · : G × G −→ G definitˇ a prin (h1 , k1 ) · (h2 , k2 ) = (h1 h2 , k1 k2 ) , (∀)(h1 , k1 ), (h2 , k2 ) ∈ G . (produsul direct al grupurilor H ¸si K) P. 6.2. Fie H ¸si K douˇ a grupuri (multiplicative), ϕ : H −→ Aut(K) un morfism de grupuri, iar G = H × K. Arˇ atat¸i cˇ a G este un grup ˆın raport cu operat¸ia binarˇa · : G × G −→ G definitˇa prin ϕ
(h1 , k1 ) · (h2 , k2 ) = (h1 h2 , (k1 )(h2 ) k2 ) , (∀)(h1 , k1 ), (h2 , k2 ) ∈ G . (produsul semidirect al grupurilor H ¸si K via morfismul ϕ)
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