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A [[Group|group]] each non-unit element of which is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p0710403.png" />-element, i.e. an element that satisfies an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p0710404.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p0710405.png" /> is a given prime number, the same for all elements of the group, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p0710406.png" /> is a natural number, in general different for each element of the group. In this sense <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p0710407.png" /> can be replaced by other symbols, such as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p0710408.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p0710409.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104010.png" />, but then their usage must be clearly specified. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104011.png" /> is a given prime number, such as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104012.png" /> one speaks of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104013.png" />-groups, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104014.png" />-groups, etc. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104015.png" />-groups are also called primary groups. A generalization of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104016.png" />-group is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104018.png" />-group (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104019.png" /> a given set of prime numbers), which is defined as a group each non-unit element of which is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104021.png" />-element, i.e. an element that satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104023.png" /> is a natural number all of whose prime divisors belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104024.png" />. Symbols less frequently employed in this sense include <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104026.png" />-group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104028.png" />-group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104030.png" />-group. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104031.png" /> is the set of all prime numbers, one often writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104033.png" /> and speaks of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104034.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104035.png" />-groups, and of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104036.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104037.png" />-elements. For a given group, a subgroup that is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104038.png" />-group (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104039.png" />-group) is known as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104041.png" />-subgroup (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104043.png" />-subgroup).
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104044.png" />-groups, and finite simple groups by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104045.png" />-groups (cf. [[#References|[1]]], Chapt. IV, V; [[#References|[2]]]). For this reason, the main interest is centred on the description of finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104046.png" />-groups using their Abelian subgroups or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104047.png" />-automorphisms.
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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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104048.png" />-groups have been studied to a lesser extent. A small number of the most important results, roughly subdivided into three parts, is given below.
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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]].
 
1) For the results concerning the solution of Burnside problems, cf. [[Burnside problem|Burnside problem]].
  
2) Locally finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104049.png" />-groups are non-simple (cf. [[#References|[3]]]).
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2) Locally finite $p$-groups are non-simple (cf. [[#References|[3]]]).
  
3) Examples illustrating the difference between the theory of finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104050.png" />-groups and the general theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104051.png" />-groups are: a) there exists a locally finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104052.png" />-group without non-trivial normal Abelian subgroups (cf. [[#References|[3]]]); b) there exists a locally finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104053.png" />-group that coincides with its [[Commutator subgroup|commutator subgroup]] (cf. [[#References|[3]]]). See also [[Group with a finiteness condition|Group with a finiteness condition]].
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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====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104054.png" />-adic representations and elliptic curves" , Benjamin  (1986)  (Translated from French)</TD></TR></table>
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<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====
 
====Comments====
The normalizer of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071040/p07104055.png" />-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]].
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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====
 
====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>
 
<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>

Latest revision as of 09:25, 27 June 2014

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)
How to Cite This Entry:
P-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=P-group&oldid=32311
This article was adapted from an original article by Yu.M. Gorchakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article