Let H be a group and . Then for a group G and .
This is true if H is a cyclic group. I want to know whether is this true for any other finite groups.
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Originally Posted by dimuk Let H be a group and . Then for a group G and . the answer is no. let be any non-characteristic normal subgroup of a group let and be the natural homomorphism. obviously
since is not charactersitic, there exists such that thus
Then how about the finite abelian groups?
Originally Posted by dimuk Then how about the finite abelian groups? If is a finite abelian group then where are cyclic. You proved this result for cyclic groups. Now it remains to prove (or disprove) that if it is true for then it is true for .
Originally Posted by dimuk Then how about the finite abelian groups? i don't know if you understood the idea in my previous post or not. as long as has a normal subgroup which is not characteristic, the claim in your problem would be false.
here's a simple example of finite abelian groups: let define by: then let and and define by: clearly and
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