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Let $G=C_{p^{a_{1}}}\times C_{p^{a_{2}}}\times...\times C_{p^{a_{t}}}$ where $a_{1}\geq a_{2}\geq...\geq a_{t}$ and $H\subseteq G^{P^n}$ for some integer $n$. Please prove if $n>a_{k}$ for some $k\in\{1,...,t\}$ then, $G$ and $\frac{G}{H}$ have equal rank.

The rank $G$ is minimal number generators of $G$.

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    Hint: $G/H$ is a direct product of $t$ groups. To see that these groups are non-trivial, assume that $C_{p^{a_k}} \subset H$ for some $k$. Why is this impossible?2012-05-16
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    What does $H\subseteq G^{P^n}$ mean? I can guess that you meant a lower-case $p$ and that this is assumed to be a prime number, but how am I to form a quotient of $G$ by a subgroup of a high Cartesian power of $G$? Do you mean the subgroup of elements with $p^n$-torsion or something like that? Also "$n>a_k$ for some $k\in\{1,...,t\}$" would seem to mean just $n>a_t$.2012-05-16
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    Marc, I think he meant $$G^{p^n}:==\{x^{p^n}\,;\,x\in G\}\,$$ , as G is an abelian group2012-05-16

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