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Mathematics
Linear and Abstract Algebra
Showing that every element in G can be written a certain way
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[QUOTE="Mr Davis 97, post: 6021811, member: 515461"] Let G be a finite group which possesses an automorphism ##\sigma## that has no nontrivial fixed points and ##\sigma ^2## is the identity map. Prove that ##G## is abelian. So there's a hint, and it tells me first to established that every element in ##G## can be written as ##x^{-1} \sigma (x)## for some ##x \in G##. I have two questions. Why does showing that ##f(x) x^{-1} \sigma (x)## is injective prove the hint, and how could I ever approach this problem without knowing the hint? [/QUOTE]
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Mathematics
Linear and Abstract Algebra
Showing that every element in G can be written a certain way
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