It is mentioned in here (last paragraph of the first page) that Dixon proved the following result: the probability that two elements of a finite nonabelian simple group commute is at most $\frac{1}{12}$. I cannot seem to find this proof anywhere online. Do you know a proof of this fact? Or do you have a reference for it? (I am actually more interested in the original proof)

  • 2
    In fact, for any $\epsilon$, there will be at most finitely many simple groups $G$ with $cp(G) \geq \epsilon$. This follows from Dixon's argument (linked below) and the fact that for any $n$ there are at most finitely many finite simple groups with an irreducible representation of dimension $n$. See e.g. – Ian Agol Sep 19 at 4:07

You can find Dixon's argument here (Google Books). (J.D..Dixon, Solution to Problem 176, Canadian Mathematical Bulletin 16 (1973), p302.)

  • 3
    Note that Dixon's argument can be simplified a little. No finite simple group can have an irreducible complex character of degree $2$. If $G$ were a simple group with such a representation, then $|G|$ would be even (since irreducible character degrees divide the group order). Then $G$ would contain an involution $t$ by Cauchy's theorem, and $t$ would be in $Z(G)$ since $t$ would be represented by $-I$ (on consideration of determinant), contrary to simplicity. – Geoff Robinson Sep 19 at 10:48

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

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