P-group

A 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. [1], Chapt. IV, V; [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.

2) Locally finite $p$-groups are non-simple (cf. [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. [3]); b) there exists a locally finite $p$-group that coincides with its commutator subgroup (cf. [3]). See also Group with a finiteness condition.

References

[1]	B. Huppert, "Endliche Gruppen" , 1 , Springer (1967)
[2]	D. Gorenstein, "Finite groups" , Harper & Row (1968)
[3]	O.Yu. Shmidt, "Selected works in mathematics" , Moscow (1959) (In Russian)
[4]	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
[5]	Itogi Nauki. Algebra, 1964 (1966) pp. 123–160
[6]	J.-P. Serre, "Abelian $l$-adic representations and elliptic curves" , Benjamin (1986) (Translated from French)

Comments

The normalizer of a $p$-subgroup (cf. 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 [a1]–[a2]. Local subgroups are also involved in modular representation theory for finite groups, cf. [a3]. Recently (1989) the restricted Burnside problem was solved by E.I. Zel'manov, cf. [a4], [a5] and Burnside problem.

References

[a1]	B. Huppert, "Finite groups" , 3 , Springer (1982)
[a2]	M. Aschbacher, "Finite group theory" , Cambridge Univ. Press (1986)
[a3]	J.L. Alperin, "Local representation theory" , Cambridge Univ. Press (1986)
[a4]	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)
[a5]	E.I. Zel'manov, "On the restricted Burnside problem" Sibirsk. Mat. Zh. , 30 : 6 (1989) pp. 68–74 (In Russian)