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Abelian Group Automorphism watch

1. When n is arbitrary but for . We consider P to be of the form , so elements of this subgroup can be viewed as comprising a vector space of dimension n over the finite field of p elements . The automorphisms of this subgroup are therefore given by the invertible linear transformations, so

has order *

I know that if you define and , then one has in particular and,

, however I think this follows from showing that has order .

*Since my textbook just says "this is easily shown to have order...", I'm a little unsettled. How can I show it explicitly? I've had a few ideas, but nothing seems tangible, unless I'm missing something important, which I probably am.
2. What vectors can the first column of an invertible matrix over F_p be?

Given the first column vector, what can the second column vector be?

etc.
3. (Original post by RichE)
What vectors can the first column of an invertible matrix over F_p be?

Given the first column vector, what can the second column vector be?

etc.
Got it now, cheers.

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Updated: December 2, 2011
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