The Sylow Subgroups of the Symmetric Groups

The aim of this paper is to give a direct approach to the study of the Sylow ^-subgroups Sn of the symmetric group of degree pn. [We assume throughout that p^2.] Many of the results are already known and are treated in a paper by Kaloujnine where he uses a particular representation by means of "reduced polynomials."1 It has seemed worth while to restate some of his results using the concept of complete product L o M of two permutation groups L, M which he and Krasner have recently emphasised.2 This elementary notion is of great importance in the theory of finite groups and it appears in the literature in different forms.3 A simple interpretation is given in §2 in terms of permutation matrices which shows the strong connection between the operation o and imprimitivity. The associative rule for o follows from the associative law of matrix multiplication and we show that Sn=C o C o • ■ ■ o C (n factors) where C is cyclic of order p; various expressions of Sm+n as Sm o Sn allow us to express Sm+n as a split extension and to investigate many properties of Sm+n by an inductive process. Since the Sylow ^-subgroups of the classical groups (general linear, symplectic, orthogonal and unitary) over a finite field with characteristic prime to p are direct products of the basic subgroups Sn=C o Co • • • o C (n factors) where C is cyclic of order pT (r^ l),4 it is hoped that the treatment here may also suggest ways in which these groups Sn may be studied. 5n+i=Co Sn and so Sn+i=.An-Sn where A" is elementary abelian of order pp". We show that all the factors of the series An>(An, Sn) > (An, Sn, Sn)> ■ • • are cyclic of order p and this leads naturally to a description of certain subgroups of 5„ in terms of partition diagrams. The main result of the paper is to show finally that the characteristic subgroups are precisely the normal partition subgroups.* This material was largely the content of Chapter 3 of my Cam-