Project III (MATH3382) 2013-14)

Project III (MATH3382) 2013-14

Commutators in groups

Alexander Stasinski

Description

Let ${G}$ be a group. For ${x,y\in G}$ , the element ${[x,y]:=x^{-1}y^{-1}xy}$ is called the commutator of ${x}$ and ${y}$ . The commutator subgroup ${[G,G]}$ is defined to be the subgroup of ${G}$ generated by the elements ${[x,y]}$ for ${x,y\in G}$ . In group theory, ${[G,G]}$ is an important subgroup which is used to define various classes of groups such as solvabe and nilpotent groups. The group ${G}$ is called perfect if ${G=[G,G]}$ .

By definition, we have ${\{[x,y]:x,y\in G\}\subset [G,G]}$ , but in general this inclusion is not an equality because there are examples of groups where the set of commutators ${\{[x,y]:x,y\in G\}}$ is different from ${[G,G]}$ .

Nevertheless, In 1951 Ito and Ore independently proved the following for the symmetric and alternating groups:

Theorem. If ${G=S_n}$ or ${G=A_n}$ we have ${\{[x,y]:x,y\in G\}=[G,G]}$ .

Since for ${G=A_n}$ we have ${G=[G,G]}$ , the above result says that any element in an alternating group is a commutator. The alternating groups are simple for n>4. Furthermore, Ore stated that “it is possible that a similar theorem holds for any simple group of finite order, but it seems that at present we do not have the necessary methods to investigate the question”. Through the years, the question of determining whether the previous theorem extends to all finite (non-abelian) simple groups has become known as the Ore conjecture.

After a series of papers by various authors, dealing with special cases, the conjecture was finally settled completely in 2010 by Liebeck, O’Brien, Shalev and Tiep:

Theorem. If ${G}$ is a non-abelian finite simple group, then every ${g\in G}$ is a commutator.

The project will involve understanding various special cases of the Ore conjecture (for example the alternating groups) or commutators in symmetric groups or finite matrix groups like ${[G,G]}$ or ${[G,G]}$ . It will then be possible to go further in various directions, such as simple groups of Lie type (a series of families of matrix groups). There are also several stimulating open problems that can be computed in special cases, for example the following recent question by Shalev:

Question. Is it true that all elements of ${[G,G]}$ , ${[G,G]}$ are commutators?

Prerequisites

Algebra II is essential.

Resources

[1] A blog post with a summary of the Ore conjecture. link

[2] Kappe and Morse, On commutators in groups. A nice survey. link

[3] Shalev, Commutators, words, conjugacy classes and character methods. Another survey with slightly broader scope. link

[4] Ore, Some remarks on commutators. The paper that started it all. link

[5] Liebeck, O'Brien, Shalev and Tiep, The Ore conjecture. The paper that finished off the proof. link

email: Alexander Stasinski

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