Anyone can give me any clue about the proof of $\text{Aut}(\mathbb{Z}/p \times\cdots \times \mathbb{Z}/p )\simeq GL_n(\mathbb{Z}/p)$? Thank you.
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5Show any automorphism is a vector-space isomorphism over $\Bbb Z/p\Bbb Z$ (which is very easy), and you are done. – PVAL-inactive Sep 05 '14 at 06:43
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See http://math.stackexchange.com/questions/805590 for the case $n=2$ – Derek Holt Sep 05 '14 at 08:18
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If $A,B$ are abelian groups with $pA=pB=0$, then $A,B$ are $\mathbb{Z}/p$-vector spaces in a unique way. Besides, a homomorphisms $A \to B$ of abelian groups is already $\mathbb{Z}/p$-linear (since it is $\mathbb{Z}$-linear and $\mathbb{Z}$ maps onto $\mathbb{Z}/p$). It follows in particular that automorphisms of $A$ as an abelian group coincide with automorphisms of $A$ as a $\mathbb{Z}/p$-vector space.
Martin Brandenburg
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