Suppose A is a finite abelian group and p is a prime. A^p={a^p : a in A} and A_p={x:x^p=1,x in A}.(adsbygoogle = window.adsbygoogle || []).push({});

How to show A/A^p is isomorphic to A_p.

I tried to define a p-power map between A/A^p and A_p and show this map is isomorphism.

But my idea didnot work right now. Please give me some help.

In addition, How to show the number of subgroups of A of order p equals the number of

subgroups of A of index p.

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# Finite abelian group

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