Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Given magma $(X, *)$ and that $(x*y)*y = x$ and $y*(y*x) = x$ $\forall x, y \in X$ prove that $x*y = y*x$.

Should be rather simple but I've been trying to prove that for several hours now with no luck. I've been trying lots of substitutions in $x*y$ and $y*x$ but wasn't able to come to something identical.

share|cite|improve this question
up vote 5 down vote accepted

This is similar to question A1 on the 2001 Putnam Exam, though the solution is a little different. You just have to play around a bit (or a lot, if you're new to this sort of thing):

Claim 1: $(x\star y)\star x=y$ for all $x,y\in X$. Proof: Let $x,y\in X$ be arbitrary. Then, $(x\star y)\star y=x$ by assumption. Now left-multiply by $(x\star y)$ to see that $(x\star y)\star((x\star y)\star y)=(x\star y)\star x$. Now apply your second property to the left hand side, with $(x\star y)$ taking the role of $y$, to see that the left hand side is actually just $y$. Thus $y=(x\star y)\star x$.

Now, since $(x\star y)\star x=y$ for all $x,y$, simply multiply on the right by $x$ to obtain $$((x\star y)\star x)\star x=y\star x$$

Use your first property with $(x\star y)$ playing the role of $x$ and $x$ playing the role of $y$, and you're done!

share|cite|improve this answer

In short: $$x*y=((x*y)*x)*x=((x*y)*((x*y)*y))*x=y*x$$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.