Ore's Conjecture for perfect groups

Question

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.

Answer 1

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.

Answer 2

The counterexample described by Derek Holt can easily be checked with GAP as follows:

gap> G := Image(IsomorphismPermGroup(PerfectGroup(960,2)));
A5 2^4'
gap> CommutatorLength(G); # > 1 => there are elements which are not commutators 
2
gap> GeneratorsOfGroup(G); # permutations generating the group
[ (1,2)(3,6)(5,8)(9,10), (2,4,5)(6,7,9), (1,3)(4,7)(5,9)(8,10),
  (1,3)(2,6)(5,9)(8,10), (1,3)(2,6)(4,7)(8,10), (2,6)(4,7)(5,9)(8,10) ]
gap> noncommutators := Difference(AsList(G),Set(Tuples(AsList(G),2),Comm));;
gap> Length(noncommutators);
120
gap> noncommutators; # the entire list of non-commutators
[ (1,2)(3,6)(4,5,7,9)(8,10), (1,2)(3,6)(4,7)(5,8,9,10),
  (1,2)(3,6)(4,7)(5,10,9,8), (1,2)(3,6)(4,8,7,10)(5,9),
  (1,2)(3,6)(4,9,7,5)(8,10), (1,2)(3,6)(4,10,7,8)(5,9),
  (1,2,3,6)(4,5)(7,9)(8,10), (1,2,3,6)(4,7)(5,8)(9,10),
  (1,2,3,6)(4,7)(5,10)(8,9), (1,2,3,6)(4,8)(5,9)(7,10),
  (1,2,3,6)(4,9)(5,7)(8,10), (1,2,3,6)(4,10)(5,9)(7,8),
  (1,3)(2,4)(5,8,9,10)(6,7), (1,3)(2,4)(5,10,9,8)(6,7),
  (1,3)(2,4,6,7)(5,8)(9,10), (1,3)(2,4,6,7)(5,10)(8,9),
  (1,3)(2,5)(4,8,7,10)(6,9), (1,3)(2,5,6,9)(4,8)(7,10),
  (1,3)(2,5)(4,10,7,8)(6,9), (1,3)(2,5,6,9)(4,10)(7,8),
  (1,3)(2,7,6,4)(5,8)(9,10), (1,3)(2,7,6,4)(5,10)(8,9),
  (1,3)(2,7)(4,6)(5,8,9,10), (1,3)(2,7)(4,6)(5,10,9,8),
  (1,3)(2,8,6,10)(4,5)(7,9), (1,3)(2,8)(4,5,7,9)(6,10),
  (1,3)(2,8)(4,9,7,5)(6,10), (1,3)(2,8,6,10)(4,9)(5,7),
  (1,3)(2,9,6,5)(4,8)(7,10), (1,3)(2,9)(4,8,7,10)(5,6),
  (1,3)(2,9,6,5)(4,10)(7,8), (1,3)(2,9)(4,10,7,8)(5,6),
  (1,3)(2,10,6,8)(4,5)(7,9), (1,3)(2,10)(4,5,7,9)(6,8),
  (1,3)(2,10)(4,9,7,5)(6,8), (1,3)(2,10,6,8)(4,9)(5,7),
  (1,4)(2,5,6,9)(3,7)(8,10), (1,4,3,7)(2,5)(6,9)(8,10),
  (1,4)(2,6)(3,7)(5,8,9,10), (1,4)(2,6)(3,7)(5,10,9,8),
  (1,4,3,7)(2,6)(5,8)(9,10), (1,4,3,7)(2,6)(5,10)(8,9),
  (1,4)(2,8,6,10)(3,7)(5,9), (1,4,3,7)(2,8)(5,9)(6,10),
  (1,4)(2,9,6,5)(3,7)(8,10), (1,4,3,7)(2,9)(5,6)(8,10),
  (1,4)(2,10,6,8)(3,7)(5,9), (1,4,3,7)(2,10)(5,9)(6,8),
  (1,5,3,9)(2,4)(6,7)(8,10), (1,5)(2,4,6,7)(3,9)(8,10),
  (1,5)(2,6)(3,9)(4,8,7,10), (1,5,3,9)(2,6)(4,8)(7,10),
  (1,5)(2,6)(3,9)(4,10,7,8), (1,5,3,9)(2,6)(4,10)(7,8),
  (1,5)(2,7,6,4)(3,9)(8,10), (1,5,3,9)(2,7)(4,6)(8,10),
  (1,5)(2,8,6,10)(3,9)(4,7), (1,5,3,9)(2,8)(4,7)(6,10),
  (1,5)(2,10,6,8)(3,9)(4,7), (1,5,3,9)(2,10)(4,7)(6,8),
  (1,6,3,2)(4,5)(7,9)(8,10), (1,6,3,2)(4,7)(5,8)(9,10),
  (1,6,3,2)(4,7)(5,10)(8,9), (1,6,3,2)(4,8)(5,9)(7,10),
  (1,6,3,2)(4,9)(5,7)(8,10), (1,6,3,2)(4,10)(5,9)(7,8),
  (1,6)(2,3)(4,5,7,9)(8,10), (1,6)(2,3)(4,7)(5,8,9,10),
  (1,6)(2,3)(4,7)(5,10,9,8), (1,6)(2,3)(4,8,7,10)(5,9),
  (1,6)(2,3)(4,9,7,5)(8,10), (1,6)(2,3)(4,10,7,8)(5,9),
  (1,7,3,4)(2,5)(6,9)(8,10), (1,7)(2,5,6,9)(3,4)(8,10),
  (1,7,3,4)(2,6)(5,8)(9,10), (1,7,3,4)(2,6)(5,10)(8,9),
  (1,7)(2,6)(3,4)(5,8,9,10), (1,7)(2,6)(3,4)(5,10,9,8),
  (1,7,3,4)(2,8)(5,9)(6,10), (1,7)(2,8,6,10)(3,4)(5,9),
  (1,7,3,4)(2,9)(5,6)(8,10), (1,7)(2,9,6,5)(3,4)(8,10),
  (1,7,3,4)(2,10)(5,9)(6,8), (1,7)(2,10,6,8)(3,4)(5,9),
  (1,8,3,10)(2,4)(5,9)(6,7), (1,8)(2,4,6,7)(3,10)(5,9),
  (1,8,3,10)(2,5)(4,7)(6,9), (1,8)(2,5,6,9)(3,10)(4,7),
  (1,8,3,10)(2,6)(4,5)(7,9), (1,8)(2,6)(3,10)(4,5,7,9),
  (1,8)(2,6)(3,10)(4,9,7,5), (1,8,3,10)(2,6)(4,9)(5,7),
  (1,8)(2,7,6,4)(3,10)(5,9), (1,8,3,10)(2,7)(4,6)(5,9),
  (1,8)(2,9,6,5)(3,10)(4,7), (1,8,3,10)(2,9)(4,7)(5,6),
  (1,9,3,5)(2,4)(6,7)(8,10), (1,9)(2,4,6,7)(3,5)(8,10),
  (1,9,3,5)(2,6)(4,8)(7,10), (1,9)(2,6)(3,5)(4,8,7,10),
  (1,9,3,5)(2,6)(4,10)(7,8), (1,9)(2,6)(3,5)(4,10,7,8),
  (1,9)(2,7,6,4)(3,5)(8,10), (1,9,3,5)(2,7)(4,6)(8,10),
  (1,9,3,5)(2,8)(4,7)(6,10), (1,9)(2,8,6,10)(3,5)(4,7),
  (1,9,3,5)(2,10)(4,7)(6,8), (1,9)(2,10,6,8)(3,5)(4,7),
  (1,10,3,8)(2,4)(5,9)(6,7), (1,10)(2,4,6,7)(3,8)(5,9),
  (1,10,3,8)(2,5)(4,7)(6,9), (1,10)(2,5,6,9)(3,8)(4,7),
  (1,10,3,8)(2,6)(4,5)(7,9), (1,10)(2,6)(3,8)(4,5,7,9),
  (1,10)(2,6)(3,8)(4,9,7,5), (1,10,3,8)(2,6)(4,9)(5,7),
  (1,10)(2,7,6,4)(3,8)(5,9), (1,10,3,8)(2,7)(4,6)(5,9),
  (1,10)(2,9,6,5)(3,8)(4,7), (1,10,3,8)(2,9)(4,7)(5,6) ]

Answer 3

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.

Answer 4

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.