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4. Suppose we are given a group H and a group N, together with a group homomorphism : H→ Aut(N). Let G be the Cartesian
Posted: Tue Jul 12, 2022 11:39 am
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- 4 Suppose We Are Given A Group H And A Group N Together With A Group Homomorphism H Aut N Let G Be The Cartesian 1 (74.17 KiB) Viewed 26 times
4. Suppose we are given a group H and a group N, together with a group homomorphism : H→ Aut(N). Let G be the Cartesian product N × H, and m be the binary operation GXG G ((n1, h₁), (n2, h₂)) → (n10(h₁) (n2), h₁h₂) (i) Show that G has the structure of a group with this binary operation. [3 marks] (ii) Let en denote the identity element of N. Show that {en} x H is a subgroup of G isomorphic to H. [2 marks] (iii) Let e denote the identity element of H. Show that N x {e} is a normal subgroup of G isomorphic to N. [2 marks]