# Proof of abelian-ness if every element is also its own inverse

by Juanriq
Tags: abelianness, element, inverse, proof
 P: 42 Salutations! I just want to make sure I am on the right track... 1. The problem statement, all variables and given/known data Let G be a group. Prove that if every element in a group is equal to its own inverse, then G is abelian 2. The attempt at a solution Pick two elements $a, b \in \thinspace G$. By assumption, $a = a^{-1}$ and $b = b^{-1}$. Our goal is to show that $ab = ba$. By our assumption, the following holds $ab = a^{-1}b^{-1}$ Since G is a group, it is closed under inverses, therefore we can take the inverse of both sides $(ab)^{-1} = \bigl (a^{-1} b^{-1} \bigl )^{-1} = \bigl (b^{-1} \bigl )^{-1} \bigl (a^{-1} \bigl )^{-1} = ba$ Thanks!
 P: 367 ab is an element of G, what does that mean (ab)^-1 is by assumption?
 P: 42 Oh! By assumption [latex] ab = (ab)^{-1} [\latex] as well. That definitely makes this a simple one-liner when I apply the inverse and initial assumpition to the right side. Thanks!

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