The Sylow subgroups of the symmetric group

Abstract

In the Sylow theorems t we learn that if the order of a group 9 is divisible by pa (p a prime integer) and not by pa+l, then W contains one and only one set of conjugate subgroups of order pa , and any subgroup of W whose order is a power of p is a subgroup of some member of this set of conjugate subgroups of W. These conjugate subgroups may be called the Sylow subgroups of W. It will be our purpose to investigate the Sylow subgroups of the symmetric group of substitutions. By means of a preliminary lemma the discussion will be reduced to the case where the degree of the symmetric group is a power (pa) of the prime (p) under consideration. A set of generators of the group having been obtained, they are found to set forth, in the notation suggested by their origin, the complete imprimitivity of the group. The various groups of substitutions upon the systems of imprimitivity, induced by the substitutions of the original group, are seen to be themselves Sylow subgroups of symmetric groups of degrees the various powers of p less than pa. They are also the quotient groups under the initial group of an important series of invariant subgroups. In terms of the given notation convenient exhibitions are obtained of the commutator series of subgroups and also of all subgroups which may be considered as the Sylow subgroups of symmetric groups of degree a power of p. Enumerations are made of the substitutions of periods p and pa and the conjugacy relations of the latter set of substitutions are discussed. The subgroup consisting of all substitutions invariant under the main group is cyclic and of order p. The full set of conjugate Sylow subgroups of the symmetric group on pa letters fall into