# Group With an Endomorphism That is “Almost” Abelian is Abelian.

Suppose a finite group has the property that for every $x, y$, it follows that

\begin{equation*} (xy)^3 = x^3 y^3. \end{equation*}

How do you prove that it is abelian?

Edit: I recall that the correct exercise needed in addition that the order of the group is not divisible by 3.

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You don't, as the group is not necessarily abelian! The group of upper triangular 3-by-3 matrices with ones along the diagonal and coefficients in the three-element field $\mathbb Z/3\mathbb Z$ has exponent three, so your equation holds, but it is not abelian. –  Mariano Suárez-Alvarez Jul 28 '10 at 21:47
(There are lots of examples: the most famous ones are the Burnside groups B(m,3), which you'll find described at en.wikipedia.org/wiki/…; the group in the first comment is B(2,3)) –  Mariano Suárez-Alvarez Jul 28 '10 at 21:49
@Mariano, why don't you give that as the answer and then it can be accepted? Otherwise it looks as though nobody has answered the question. –  bryn Jul 29 '10 at 6:57
By the way, your statement becomes true if you change 3 by 2. –  falagar Jul 29 '10 at 14:54

On the other hand, if the order of your group is not a multiple of 3 then it must be abelian!

You can read a proof here

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Urgh! Why do they write proofs in such a complicated way?! –  Mariano Suárez-Alvarez Jul 29 '10 at 16:25
Yeah! This is what I had in mind! Thanks! :) –  user218 Jul 29 '10 at 18:07

I wrote a short proof here .

Steve

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You don't, as the group is not necessarily abelian! The group of upper triangular 3-by-3 matrices with ones along the diagonal and coefficients in the three-element field $\mathbb {Z}/3\mathbb{Z}$ has exponent three, so your equation holds, but it is not abelian.

There are lots of examples: the most famous ones are the Burnside groups $B(m,3)$: the group I described above is $B(2,3)$.

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It would be instructive to know what motivated the downvote... –  Mariano Suárez-Alvarez Jul 29 '10 at 18:10
It was because I was annoyed that you started a meta thread on me unnecessarily. If you have had objections then you could have left them as comments and I would have read them. –  user218 Jul 29 '10 at 18:16
@Line Bundle, you downvoted this answer because I asked, prompted by another question, about policy? –  Mariano Suárez-Alvarez Jul 29 '10 at 18:20
Yes, and I later regretted and tried removing the vote; but here one apparently can't do it. If you'll edit your question I can remove it. –  user218 Jul 29 '10 at 18:21
Don't worry about it! –  Mariano Suárez-Alvarez Jul 29 '10 at 18:24