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Published online by Cambridge University Press: 20 April 2010
Abstract
Commutators originated over 100 years ago as a by-product of computing group characters of nonabelian groups. They are now an established and immensely useful tool in all of group theory. Commutators became objects of interest in their own right soon after their introduction. In particular, the phenomenon that the set of commutators does not necessarily form a subgroup has been well documented with various kinds of examples. Many of the early results have been forgotten and were rediscovered over the years. In this paper we give a historical overview of the origins of commutators and a survey of different kinds of groups where the set of commutators does not equal the commutator subgroup. We conclude with a status report on what is now called the Ore Conjecture stating that every element in a finite nonabelian simple group is a commutator.
Origins of commutators
“In a group the product of two commutators need not be a commutator, consequently the commutator group of a given group cannot be defined as the set of all commutators, but only as the group generated by these. There seems to exist very little in the way of criteria or investigations on the question when all elements of the commutator group are commutators.”
This is what Oystein Ore says in 1951 in the introduction to his paper “Some remarks on commutators”. Since Ore made his comments, numerous contributions have been made to this topic and they are widely scattered over the literature.
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