How to prove abelian group

Question

I am struggling answering this question for myself: How can i prove that a Group $G$ is abelian, if $$g\circ g=e \ \forall g \in G $$

A group is abelian if this is true: $$a\circ b = b\circ a\ \forall a,b \in G$$ But i dont understand how to prove this.

Hope someone can help me out with this!

Simplify $ababba$ in two different ways: one way by replacing $bb$ by $e$ as your first step, another by replacing $abab=(ab)(ab)$ by $e$ as your first step. — Akiva Weinberger, Feb 21 '16 at 19:06

score 4 · Accepted Answer · answered Feb 21 '16 at 18:49

4

$a$ and $b$ commute iff $a \circ b = b \circ a$ iff $a \circ b \circ a^{-1} \circ b ^{-1} = e$.

However $a^{-1} = a$ and $b^{-1} = b$ so then $a \circ b \circ a^{-1} \circ b ^{-1} = a \circ b \circ a \circ b = (a \circ b) \circ (a \circ b) = e$ by hypothesis.

answered Feb 21 '16 at 18:49

James

5,443

How does $a=a^{-1}$ and $b=b^{-1}$ without assuming commutativity? – Michael T Chase Apr 29 '21 at 21:08
Nevermind: https://math.stackexchange.com/questions/3115411/when-is-a-matrix-equal-to-its-own-inverse – Michael T Chase Apr 29 '21 at 21:12

score 2 · Answer 2 · answered Feb 21 '16 at 18:49

2

Note that $g=g^{-1}$ for any $g$. So, $(ab)^2=abab=e$ and thus $ab=b^{-1}a^{-1} = ba$ for all $a,b \in G$.

answered Feb 21 '16 at 18:49

sqtrat

2,257

How to prove abelian group

2 Answers2