Suppose $G$ $G$ is a group where all non-identity elements are of order 2. Is $G$ $G$ abelian ?

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Answer 1 · 2018-02-16T18:27:13

Yes

Let $a, b \in G$ $a, b in G$

Then:

$a b = b^{2} a b a^{2} = (b b) a b (a a) = b (b a b a) a = b {(b a)}^{2} a = b a$ $ab = b^2 ab a^2 = (b b) ab (a a) = b(ba ba)a = b (ba)^2 a = ba$

Suppose GG is a group where all non-identity elements are of order 2. Is GG abelian ?