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Øystein Ore proved in 1951 that for $n\gt4$ we have that all elements of $A_n$ are commutators. He subsequently conjectured that this is true for all finite non-abelian simple groups.

In this connection, not long ago I asked if every element of a perfect group is a commutator. It was answered in the negative. I should have caught that $A_5*A_5$ is a counterexample. The smallest such was found to be of order 960.

Ore's conjecture/theorem was proved in 2008, and relies on the classification theorem.

• See M. W. Libeck, E. A. O'Brien, Aner Shalev, Pham Huu Tiep, The Ore conjecture, Journal of the European Mathematical Society, vol. 12, issue 4, 2010. pp. 939-1008. http://doi.org/10.4171/JEMS/220

Interestingly Ore himself was mainly concerned with graph theory.

Øystein Ore proved in 1951 that for $n\gt4$ we have that all elements of $A_n$ are commutators. He subsequently conjectured that this is true for all finite non-abelian simple groups.

In this connection, not long ago I asked if every element of a perfect group is a commutator. It was answered in the negative. I might have caught that $A_5*A_5$ is a counterexample. The smallest such was found to be of order 960.

Ore's conjecture/theorem was proved in 2008, and relies on the classification theorem.

• See M. W. Libeck, E. A. O'Brien, Aner Shalev, Pham Huu Tiep, The Ore conjecture, Journal of the European Mathematical Society, vol. 12, issue 4, 2010. pp. 939-1008. http://doi.org/10.4171/JEMS/220

Interestingly Ore himself was mainly concerned with graph theory.

Øystein Ore proved in $1951$ that for $n\gt4$ we have that all elements of $A_n$ are commutators. He subsequently conjectured that this is true for all finite non-abelian simple groups.

In this connection, not long ago I asked if every element of a perfect group is a commutator. It was answered in the negative. I should have caught that $A_5*A_5$ is a counterexample. The smallest such was found to be of order $960.$

Ore's conjecture/theorem was proved in $2008,$ and relies on the classification theorem.

• See M. W. Libeck, E. A. O'Brien, Aner Shalev, Pham Huu Tiep, Journal of the European Mathematical Society, vol. $12$, issue $4, 2010.$ pp. $939-1008.$

Interestingly Ore himself was mainly concerned with graph theory.

Øystein Ore proved in 1951 that for $n\gt4$ we have that all elements of $A_n$ are commutators. He subsequently conjectured that this is true for all finite non-abelian simple groups.

In this connection, not long ago I asked if every element of a perfect group is a commutator. It was answered in the negative. I should have caught that $A_5*A_5$ is a counterexample. The smallest such was found to be of order 960.

Ore's conjecture/theorem was proved in 2008, and relies on the classification theorem.

• See M. W. Libeck, E. A. O'Brien, Aner Shalev, Pham Huu Tiep, The Ore conjecture, Journal of the European Mathematical Society, vol. 12, issue 4, 2010. pp. 939-1008. http://doi.org/10.4171/JEMS/220

Interestingly Ore himself was mainly concerned with graph theory.

Øystein Ore proved in $1951$ that for $n\gt4$ we have that all elements of $A_n$ are commutators. He subsequently conjectured that this is true for all finite non-abelian simple groups.

In this connection, not long ago I asked if every element of a perfect group is a commutator. It was answered in the negative. I should have caught that $A_5*A_5$ is a counterexample. The smallest such was found to be of order $960.$

Ore's theorem was proved in $2008,$ and relies on the classification theorem.

• See M. W. Libeck, E. A. O'Brien, Aner Shalev, Pham Huu Tiep, Journal of the European Mathematical Society, vol. $12$, issue $4, 2010.$ pp. $939-1008.$

Interestingly Ore himself was mainly concerned with graph theory.

Øystein Ore proved in $1951$ that for $n\gt4$ we have that all elements of $A_n$ are commutators. He subsequently conjectured that this is true for all finite non-abelian simple groups.

In this connection, not long ago I asked if every element of a perfect group is a commutator. It was answered in the negative. I should have caught that $A_5*A_5$ is a counterexample. The smallest such was found to be of order $960.$

Ore's conjecture/theorem was proved in $2008,$ and relies on the classification theorem.

• See M. W. Libeck, E. A. O'Brien, Aner Shalev, Pham Huu Tiep, Journal of the European Mathematical Society, vol. $12$, issue $4, 2010.$ pp. $939-1008.$

Interestingly Ore himself was mainly concerned with graph theory.