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why if G is an abelian p-group not divisible then exists an element g in G which is not divisible by p? thanks

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closed as off topic by Qiaochu Yuan, Pete L. Clark, Mark Sapir, Ryan Budney, Andreas Blass Jun 15 '11 at 22:27

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Because any element is divisible by any other prime. –  Pace Nielsen Jun 15 '11 at 19:38

1 Answer 1

up vote 0 down vote accepted

As Pace said, but with more detail:

If $G$ is an abelian $p$-group, then for any $g$ in $G$, the order of $g$ is a power of $p$, say $p^k$. Thus for any integer $n$ coprime with $p$, $n$ is a unit (mod $p^k$), so for some $m$, $nm=1$ mod $p^k$. So $n(mg)=(nm)g=(ap^k+1)g=g+a(p^kg)=g+0=g$. Thus $g$ is divisible by $n$.

This holds for any $g$; so if every $g$ is divisible by $p$, they are also divisible by $p^n$ for all $n$, so they are all divisible by $p^nk$ for any $n$ and any $k$ coprime to $n$, which is to say, any nonzero number.

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