# Classification theorem for specific non-Abelian groups

I recently learned about the fundamental theorem of finite Abelian groups, that any finitely generated commutative group $G$is isomorphic to a direct sum: $\mathbb Z^n \displaystyle \oplus \mathbb Z_{j_1} \oplus \mathbb Z_{j_2} \oplus...\oplus\mathbb Z_{j_k}$ where the subscripts are prime powers.

What about a noncommutative group where, in particular, the operation $*$ is such that $x*y= (y*x)^{-1}$? Does a similar classification theorem exist in that case? I asked my professor about it and he mentioned something about 2-torsion groups that went over my head.

• Sorry I meant that $(y*x) = (x*y)^{-1}$. I will make an edit. – KR136 Mar 5 '17 at 22:40

## 1 Answer

Any group $G$ in which $xy=(yx)^{-1}$ for all $x,y\in G$ is automatically abelian. To prove this, notice that if you let $y$ be the identity, this says $x=x^{-1}$ for any $x$. Applying this with $yx$ in place of $x$ then gives $yx=(yx)^{-1}=xy$. That is, $G$ is abelian.