Let G be a abelian group with . Prove that G areisomorphic with

End(G) ={f:G->G with f(xy)=f(x)f(y) } . If G is abelian then End(G) is group .

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- Jan 27th 2009, 05:58 AM #1

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- Jan 27th 2009, 04:26 PM #2

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i think you should change the title of your post to "|End(G)|=|G|", because right now it has nothing to do with your post! anyway, this is a good question, especially for those who would like to

see non-trivial applications of fundamental theorem of finite abelian groups. to solve the problem,__i'll assume that____is not cyclic__and i'll show that :

we have where and let for any

define by: it's easy to see that and if and only if

therefore: the only thing left to prove is that: so we define by: note that

is well-defined because it's clear that now suppose that for some then for some which is obviously nonsense!