- #1

fishturtle1

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## Homework Statement

Prove in any finite group G, the number of elements not equal to their own inverse is an even number.

## Homework Equations

if ab = ba = e, then a = b

^{-1}and b = a

^{-1}

## The Attempt at a Solution

Let S, A, B, be subsets of G where S = A + B.

Let a ∈ A s.t. there exists a unique b ∈ B so that

ab = ba = e and a =/= b.

Then |A| = |B| = k, k ∈ ℤ.

Then |S| = |A| + |B| = k + k = 2k.

And the definition of even is 2k, so |S| will always be even. []

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