Let G be a group. A group isomorphism from G to G is called a group...

  

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Let G be a group. A group isomorphism from G to G is called a group automorphism of G. Let Aut(G) denote the set of all group automorphism of G. Then Aut(G) form a subgroup of Sym(G). Let G ba a group and K is a subgroup of G contained in Z(G). (a) Show that K is a normal subgroup of G. (b) Suppose that G/K is cyclic. Show that G is abelian. Let G be a group. A group isomorphism from G to G is called a group automorphism of G. Let Aut(G) denote the set of all group automorphism of G. Then Aut(G) form a subgroup of Sym(G). Let G ba a group and K is a subgroup of G contained in Z(G). (a) Show that K is a normal subgroup of G. (b) Suppose that G/K is cyclic. Show that G is abelian.

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