Proof of Infinite Cyclic Group Isomorphism to Z

[radou]

Any map is surjective onto its image.

So if I look at the epimorphism f : G --> Im(f), I could apply the corollary? eg, since Im(f) and every other subgroup of H is normal, there must exist a normal subgroup of H to which N is mapped?

 

[matt grime]

Forget H. Replace H with Im(f).

 

[radou]

OK, so f : G --> Im(f) is an epimorphism. Since there exists a bijection between the set of all subgroups of G which contain Ker(f) and the set of all subgroups of Im(f) (where normal subgroups correspond to normal ones), the group N < G is mapped to some subgroup K < Im(f). Since K is normal (because it is a subgroup of an abelian group), N must be normal in G.

 

[fawad]

can some body help me in solving the following problem.
a finite group with an even number of elements contains an even number of elements x such that x^-1 = x.