# Sylow subgroup

A maximal $\pi$-subgroup of a group, where $\pi$ is a set of prime numbers; that is, a periodic subgroup whose elements have orders that are divisible only by the prime numbers from $\pi$ and which is not contained in any larger subgroup with this property (a Sylow $\pi$-subgroup). The Sylow $p$-subgroups, that is, those for which $\pi$ consists of one prime number $p$, have fundamental significance in group theory. The name is given in honour of L. Sylow, who proved a number of theorems on such subgroups in a finite group (see Sylow theorems).
Sylow subgroups play a major role in the theory of finite groups. Thus, the question of complementation of a normal Abelian subgroup reduces to the same question for Sylow subgroups; the existence of non-trivial $p$-quotient groups is connected with the existence of non-trivial $p$-quotient groups for the normalizer of a Sylow $p$-subgroup; the structure of a finite simple group is largely determined by the structure of its Sylow $2$-subgroups. In the theory of infinite groups, except in the theory of locally finite groups, the role of Sylow subgroups is less important, since the fundamental question of conjugacy of Sylow $p$-subgroups no longer has a positive solution, except in special cases.