- #1

Mr Davis 97

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## Homework Statement

Let ##p## be a prime integer. Show that ##\operatorname{Aut}(\underbrace{Z_p\times \dots \times Z_p}_{n \text{ factors }})\cong GL_n(\mathbb{F}_p)##, where ##\mathbb{F}_p## is ##Z_p## viewed as a field.

## Homework Equations

## The Attempt at a Solution

First, note that ##Z_p = \{1,x,\dots,x^{p-1}\}##.= is the cyclic group of order ##p##. Also, let ##e_i## be the element in ##(Z_p)^n## with ##x## in the ##i##-th position and the identity everywhere else.

Here is my idea of what an isomorphism would look like:

##\Psi : GL_n(\mathbb{F}_p) \to \operatorname{Aut}((Z_p)^n)## given by ##(v_1~\dots~v_n) \mapsto \alpha## where ##\alpha(e_i) = v_i##.

So here are my questions. How can I show that this map is well-defined in the sense that the ##\alpha## being mapped to is actually an automorphism? Also, how would I show that this map is onto?