# Minimal generation of simple groups and Ore's conjecture

The well known Ore's conjecture (now established) states that every element of a finite non-abelian simple group $$G$$ is a commutator of a pair of elements. Also we know that $$G$$ is $$2$$-generated.

I am trying to find out what is known about: given any $$1 \neq x \in G$$, can it be a commutator of two generating elements, i.e., $$x = [a,b]$$ so that $$G$$ is generated by $$a, b$$ as well.

If the answer is negative, are there known restrictions on the conjugacy class of $$x$$ for which this happens?

The question is motivated from the action of $$G$$ on Riemann surfaces that yield orbit genus $$1$$ corresponding to minimal signatures for the group.

• This has been discussed elsewhere, and there are lots of counterexamples, the smallest of which is $A_5$. In $A_5$, if $[a,b]$ is an element of order $2$, then $a$ and $b$ both stabilize the same point and generate a subgroup $A_4$. – Derek Holt May 21 at 22:10
• Thanks for pointing it out. My prime concern was while experimenting on action of $PSL(2,p)$ on Riemann surfaces with orbit genus $1$, where $p$ is a prime. It now seems like "almost often" (except the class $A_n$ and a few other low order cases), other than order $2$ element, rest of them are commutators with its components generating the group $G$. I need some time to close the question. – Siddhartha May 21 at 22:58

This question was answered on another forum, so I will just repeat the answer from there.

It is true for 'most' finite simple groups, but there are lots of exceptions, including $${\rm PSL}(2,2^n)$$ for all $$n$$, $${\rm PSL}(3,3)$$, $${\rm PSU}(3,3)$$, $$A_8$$, $${\rm PSp}(4,3)$$, and $$M_{11}$$. It is not true in general in $${\rm SL}(n,q)$$ and $${\rm Sp}(2n,q)$$, so they are also exceptions whenever they have trivial centre.

In particular, $$A_5$$ is an exception. If $$a,b \in A_5$$ with $$[a,b]$$ of order $$2$$, then $$\langle a,b \rangle \cong A_4$$.

Here is a character-theoretic argument which shows that for each $$n > 1$$, whenever an involution $$t \in {\rm SL}(2,2^{n})= G$$ has $$t = [a,b]$$, then $$a,b \in B = N_{G}(S)$$, where $$S$$ is the unique Sylow $$2$$-subgroup of $$G$$ containing $$t$$.

In general, when $$G$$ is a finite group and $$x \in G$$, then the number of ordered pairs $$(a,b) \in G \times G$$ with $$x = [a,b]$$ is expressible as $$\sum_{ \chi \in {\rm Irr}(G)} \frac{|G|\chi(x)}{\chi(1)},$$ where $${\rm Irr}(G)$$ is the set of complex irreducible characters of $$G$$. This formula was probably known to W. Burnside (in fact, the fact that $$x$$ is a commutator if and only if the sum is positive appears in Burnside's book, and was later important in the proof of the Ore conjecture).

Letting $$T,S, B, G$$ be as above, we note that $$B$$ is a Frobenius group of order $$2^{n}(2^{n}-1)$$ , and has $$2^{n}-1$$ irreducible characters of degree $$1$$ (each with $$t$$ in their kernel), and one irreducible character of degree $$2^{n}-1$$ taking value $$-1$$ at $$t$$. Hence the number of order pairs $$(a,b) \in B \times B$$ with $$t = [a,b]$$ is $$(2^{n}(2^{n}-1) [ (2^{n}-1) - \frac{1}{2^{n}-1}] = 2^{3n}-2^{2n+1}$$.

On the other hand, $$G$$ has one irreducible character of degree $$1$$, the trivial character, one irreducible character of degree $$2^{n}$$ (which vanishes at $$t$$), $$2^{n-1}$$ irreducible characters of degree $$2^{n}-1$$, all taking value $$-1$$ at $$t$$, and $$2^{n-1}-1$$ irreducible characters of degree $$2^{n}+1$$, all taking value $$1$$ at $$t$$.

Hence the number of ordered pairs $$(a,b) \in G \times G$$ with $$t = [a,b]$$ is

$$2^{n}(2^{2n}-1) [ 1 - \frac{2^{n-1}}{(2^{n}-1)} + \frac{2^{n-1}-1}{2^{n}+1}],$$ which also turns out to be $$2^{3n} - 2^{2n+1}$$.

Hence all ordered pairs $$(a,b) \in G \times G$$ with $$t = [a,b]$$ actually lie in $$B \times B$$, so it is not possible to express $$t = [a,b]$$ where $$\langle a,b \rangle = G.$$