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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 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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.I. Kargapolov,   J.I. [Yu.I. Merzlyakov] Merzljakov,   "Fundamentals of the theory of groups" , Springer  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.A. Shemetkov,   "Two directions in the development of the theory of non-simple finite groups"  ''Russian Math. Surveys'' , '''30''' :  2  (1975)  pp. 185–206  ''Uspekhi Mat. Nauk'' , '''30''' :  2  (1975)  pp. 179–198</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Suzuki,   "Group theory" , '''1''' , Springer  (1982)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B. Huppert,   "Endliche Gruppen" , '''1''' , Springer  (1974)</TD></TR></table>

[[Category:Group theory and generalizations]]