Another Look at Sylow's Third Theorem

Abstract

This result is among the arsenal of tools that every first year algebra student obtains. A group where the order of every element is a power of p is called a p-group; a p-Sylow subgroup of G is a p-subgroup of G of maximal order The idea of Sylow's proof, which was originally stated in terms of permutation groups, is to look at the size of the equivalence classes obtained when all p-Sylow subgroups of G are conjugated by the elements of a fixed p-Sylow subgroup of G. The existence of a p-Sylow subgroup was needed for the proof of the third Sylow theorem, although the conclusion of the theorem certainly implies that there are p-Sylow subgroups. Using the Sylow results, Frobenius, in 1895 [1], proved a generalization: The number of subgroups of G of order ps is congruent to 1 modulo p whenever 1 < s < n. Most current texts show the existence of a p-Sylow subgroup and prove Sylow's third theorem using arguments that involve a group acting on a set. This method of proof for the existence of a p-Sylow subgroup was due to Miller between 1910 and 1915 [8, 9], but, according to Jacobson [9, p. 83], was forgotten until it was rediscovered in 1959 by Wielandt [14]. Krull [7] showed how Wielandt's method could be used to obtain Frobenius' generalization, and Gallagher, in 1967 [2], simplified the argument to one that depends upon the order of G rather than the group itself. Illustrating the combinatorics of finite group actions is part of the motivation to use this method of proof both to demonstrate the existence of p-Sylow subgroups in a finite group and to determine their number. In this note we offer another method to prove these results. Our combinatorial tool will be Mobius inversion on the lattice of subgroups of a finite group. We will see that an application of this method will easily lead to Frobenius' theorem, in fact, a generalization of it. Of course, part of the reason for presenting this proof is to highlight the method.