View source for P-group

{{TEX|done}}
A [[Group|group]] each non-unit element of which is a $p$-element, i.e. an element that satisfies an equation $x^{p^n}=1$; here $p$ is a given prime number, the same for all elements of the group, while $n$ is a natural number, in general different for each element of the group. In this sense $p$ can be replaced by other symbols, such as $q$, $r$ or $s$, but then their usage must be clearly specified. If $p$ is a given prime number, such as $2,3,5,\ldots,$ one speaks of $2$-groups, $3$-groups, etc. $p$-groups are also called primary groups. A generalization of a $p$-group is a $\pi$-group ($\pi$ a given set of prime numbers), which is defined as a group each non-unit element of which is a $\pi$-element, i.e. an element that satisfies the condition $x^m=1$, where $m$ is a natural number all of whose prime divisors belong to $\pi$. Symbols less frequently employed in this sense include $\Pi$-group, $\sigma$-group and $\tau$-group. If $N$ is the set of all prime numbers, one often writes $p'=N\setminus p$, $\pi'=N\setminus\pi$ and speaks of $p'$- and $\pi'$-groups, and of $p'$- and $\pi'$-elements. For a given group, a subgroup that is a $p$-group ($\pi$-group) is known as $p$-subgroup ($\pi$-subgroup).

Many studies in the theory of finite groups are connected with the task of describing arbitrary finite groups using finite $p$-groups, and finite simple groups by $2$-groups (cf. [[#References|[1]]], Chapt. IV, V; [[#References|[2]]]). For this reason, the main interest is centred on the description of finite $p$-groups using their Abelian subgroups or $p$-automorphisms.

Infinite (non-Abelian) $p$-groups have been studied to a lesser extent. A small number of the most important results, roughly subdivided into three parts, is given below.

1) For the results concerning the solution of Burnside problems, cf. [[Burnside problem|Burnside problem]].

2) Locally finite $p$-groups are non-simple (cf. [[#References|[3]]]).

3) Examples illustrating the difference between the theory of finite $p$-groups and the general theory of $p$-groups are: a) there exists a locally finite $p$-group without non-trivial normal Abelian subgroups (cf. [[#References|[3]]]); b) there exists a locally finite $p$-group that coincides with its [[Commutator subgroup|commutator subgroup]] (cf. [[#References|[3]]]). See also [[Group with a finiteness condition|Group with a finiteness condition]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Huppert,   "Endliche Gruppen" , '''1''' , Springer  (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D. Gorenstein,   "Finite groups" , Harper &amp; Row  (1968)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O.Yu. Shmidt,   "Selected works in mathematics" , Moscow  (1959)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.N. Chernikov,   "Finiteness conditions in the general theory of groups"  ''Transl. Amer. Math. Soc. (2)'' , '''84'''  (1969)  pp. 1–65  ''Uspekhi Mat. Nauk'' , '''14''' :  5  (1959)  pp. 45–96</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  ''Itogi Nauki. Algebra, 1964''  (1966)  pp. 123–160</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J.-P. Serre,   "Abelian $l$-adic representations and elliptic curves" , Benjamin  (1986)  (Translated from French)</TD></TR></table>



====Comments====
The normalizer of a $p$-subgroup (cf. [[Normalizer of a subset|Normalizer of a subset]]) is called a local subgroup. The study of finite simple groups heavily depends on the structure theory for their local subgroups, see [[#References|[a1]]]–[[#References|[a2]]]. Local subgroups are also involved in modular representation theory for finite groups, cf. [[#References|[a3]]]. Recently (1989) the restricted Burnside problem was solved by E.I. Zel'manov, cf. [[#References|[a4]]], [[#References|[a5]]] and [[Burnside problem|Burnside problem]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Huppert,   "Finite groups" , '''3''' , Springer  (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Aschbacher,   "Finite group theory" , Cambridge Univ. Press  (1986)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.L. Alperin,   "Local representation theory" , Cambridge Univ. Press  (1986)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E.I. Zel'manov,   "Solution of the restricted Burnside problem for groups of odd exponent"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''54''' :  1  (1990)  pp. 42–59  (In Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E.I. Zel'manov,   "On the restricted Burnside problem"  ''Sibirsk. Mat. Zh.'' , '''30''' :  6  (1989)  pp. 68–74  (In Russian)</TD></TR></table>