$G$ a finite group $n$-abelian goup and g.c.d.$\big(|G|,n(n-1)\big)=1$ , then to show $G$ is abelian

Asked Sep 14 '14 at 12:59

Active Sep 14 '14 at 12:59

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Let $G$ be a finite group and $n$ be a given positive integer such that $(ab)^n=a^nb^n , \forall a,b \in G$ and

g.c.d.$\big(|G|,n(n-1)\big)=1$ , then how to prove that $G$ is abelian ? If I can show that

$(ab)^{n(n-1)}=a^{n(n-1)}b^{n(n-1)} , \forall a,b \in G$ then I would be done , so I'm also asking , is this identity true ?

asked Sep 14 '14 at 12:59

Souvik Dey

@QuangHoang: the linked question does not answer my second question !! – Souvik Dey Sep 14 '14 at 13:13
2

Well, it doesn't matter now as the group is proved to be abelian, does it :-) – Quang Hoang Sep 14 '14 at 13:20

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