I know that any group satisfying $x^2=1$ for all $x$ is abelian. Is the same true if $x^3=1$? I don't think it is, but I can't find a basic counterexample.
If a group satisfies $x^3=1$ for all $x$, is it necessarily abelian?
15
$\begingroup$
group-theory
-
2In short, no; the result does not hold for *any* $n$ other than $n=1$ and $n=2$. – 2012-05-21
-
0It is however true if there are no elements of order 3! – 2012-05-21
-
1@Steve D: You mean, if $x^3=1$ for all $x$ and $G$ has no elements of order $3$? Well... it's a rather singular situation, don't you think? (-: – 2012-05-21
-
0@ArturoMagidin: hahaha, it's funny that what I wrote is still, somehow, correct. I was thinking of $(xy)^3=x^3y^3$. – 2012-05-21
1 Answers
19
For any odd prime $p$, there is a nonabelian group $H_p$ of order $p^3$ and such that $x^p = 1$ for all $x \in H_p$: the Heisenberg group modulo p.
Added: As Dylan Moreland points out, this expository note of Keith Conrad gives a very nice discussion of the groups of order $p^3$, including the Heisenberg groups $H_p$.
-
1See also [this handout](http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/groupsp3.pdf) of Keith Conrad's. – 2012-05-21
-
0Thanks, I'll read this stuff. – 2012-05-21