Ore's Conjecture for perfect groups We know that Ore's Conjecture is a theorem now. Does anyone know a counterexample to Ore's conjecture for perfect groups? I believe there should be one, but I dont have a counterexample. Thanks.
 A: An old paper of mine (Amer. Math. Monthly 1977) gives an easy way to
construct groups where not every element of the derived subgroup is
a commutator. In particular, if $U$ is a large enough abelian group
and $H$ is a simple group, then the derived subgroup of the wreath
product of $U$ by $H$ is perfect and contains non-commutators.
A: The counterexample referred to in the comment by NAME_IN_CAPS is the split extension of the so-called deleted permutation module for $A_5$ over ${\mathbb F}_2$. that is, the $4$-dimensional irreducible component of the $5$-dimensional permutation module. This group has $960$ elements, only $840$ of which are commutators.
Here is a general recipe for constructing examples of perfect groups in which, for any given $k$, there are elements of $[G,G]$ that are not product of at most $k$ commutators.
It can illustrated by using the other $4$-dimensional irreducible module $M$ for $A_5$ over ${\mathbb F}_2$, namely the non-absolutely irreducible module arising from the isomorphism $A_5 \cong {\rm SL}(2,4)$.
Calculations show that the tensor product $M \otimes M$ maps onto $I \oplus I$, where $I$ is the trivial module. So, for any $n \ge 1$, the semidirect product $H_n$ of $M^n$ by $A_5$ has a perfect central extension $G_n$ with centre elementary abelian of order $2^{n(n-1)/2}$, because we can get a $2^2$ for each pair of direct factors of $M^n$.
(The full Schur Multiplier of $H_n$ is larger and has the structure $2 \times 4^{2n} \times 2^{n(n-1)/2}$.)
Now $|H_n| = 2^{4n} \times 60$ and $|G_n| = 2^{n(n-1)/2} \times |H_n|$. Since a commutaor $[x,y]$ is not changed by multiplying $x,y$ by central elements, $G_n$ has at most $|H|^2$ distinct commutators, and there are at most $|H|^{2k}$ elements of $G_n$ that are products of at most $k$ commutators. This is less than $|G_n|$ for large enough $n$.
Of course, we can perform similar constructions for other modules of other simple groups.
A: There are even quasi-simple group (albeit very few) which contain non-commutators: Liebeck, O'Brien and Tiep have proved that covering groups of $A_{6}, A_{7}, L_{3}(4)$ and $U_{4}(3)$ (and of no other non-Abelian simple groups) contain non-commutators.
