A cyclic group proof

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.

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matt grime Recognitions: Homework Help Science Advisor	A cyclic group proof Forget H. Replace H with Im(f).

radou Recognitions: Homework Help	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.