Proceedings of the American Mathematical Society · 1951 · 147 citations · 1 references
Natural SciencesMathematicsAlgebraCombinatoricsSet TheoryAbstract AlgebraGroup TheoryInfinite SetsCommutationsCommutatorCommutator SubgroupCommutator TheoryDisjoint SetsInfinite GroupsPermutation Groups
In a group, the product of two commutators need not be a commutator, so the commutator subgroup is defined as the group generated by commutators, yet few criteria exist for when all elements of the commutator subgroup are themselves commutators. The study aims to demonstrate that in the finite symmetric group \(S_{2n}\) every element of the alternating group \(A_{2n}\) is a commutator, and to extend this result to \(n>5\). The authors first observe that their arguments apply to both finite and infinite cases, and note that any bijection of a set can.
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. In the following it is shown that in the finite symmetric group 2n all elements of the alternating group are commutators; one can extend this and show that when n >5 all elements in the alternating group are commutators of elements in An. For the infinite symmetric group the situation is different since we obtain: Any one-to-one correspondence of an infinite set to itself is a commutator. 1. We shall begin by making a few general remarks which apply both to the finite and the infinite cases. Any one-to-one correspondence T of an arbitrary set S to itself can be written uniquely as a product of cycles which operate on disjoint sets of elements
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