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Lie groups and Lie algebras - Exercises 1. Prove that Z(GLn (K)) = K × In . 2. Let G and H be groups and ϕ : G → H be a group homomorphism. Prove that ϕ(eG ) = eH and ϕ(a−1 ) = ϕ(a)−1 for all a ∈ G. 3. Let G, H be groups and ϕ : G → H be a group homomorphism. Prove that ker(ϕ) is a subgroup of G and im(ϕ) is a subgroup of H. 4. Let G be a group and g ∈ G. Prove that the map Ig : G → G , a 7→ gag −1 is an automorphism of G and that the map Φ : G → Aut(G) , g 7→ Ig is a group homomorphism. Prove that Inn(G) = {Ig : g ∈ G} is a normal subgroup of Aut(G). 5. Let N and G be groups and δ : G → Aut(N ) be a homomorphism. Prove that the Cartesian product N × G equipped with the operation (n1 , g1 ) ◦ (n2 , g2 ) = (n1 ◦N δ(g1 )n2 , g1 ◦G g2 ) is a group (a so-called semidirect product of N and G and denoted by N oδ G). Prove that the subgroup {(n, eG ) : n ∈ N } of N oδ G is isomorphic to N and a normal subgroup of N oδ G.

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