- #1

sheepishlion

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## Homework Statement

Prove that in any group, an element and its inverse have the same order.

## Homework Equations

none

## The Attempt at a Solution

I know that I need to show |g|=|g^-1|. This is how I approached it but I'm not sure I'm right, it seems like I'm missing something.

Let |g|=m and |g^-1|=n for g, g^-1 elements of G. Then (g)^m=e and (g^-1)^n=e. Therefore, either (g)^m <= (g^-1)^n or (g)^m >= (g^-1)^n. If (g)^m <= (g^-1)^n then e<=e and m<=n. If (g)^m >= (g^-1)^n then e>=e and m >= n. Therefore m = n and |g|=|g^-1|.

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