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单词 Pgroup
释义

p-group


Primary groups

Let p be a prime numberMathworldPlanetmath.A p-group (or p-primary group) is a group in which the order of every element is a power of p.A group that is a p-group for some prime p is also called a primary group.

Using Lagrange’s Theorem and Cauchy’s Theorem one may show that a finite groupMathworldPlanetmath G is a p-group if and only if |G| is a power of p.

Primary subgroups

A p-subgroupMathworldPlanetmathPlanetmath (or p-primary subgroup) of a group G is a subgroup (http://planetmath.org/Subgroup) H of G such that H is also a p-group.A group that is a p-subgroup for some prime p is also called a primary subgroup.

It follows from Zorn’s Lemma that every group has a maximal p-subgroup, for every prime p. The maximal p-subgroup need not be unique (though for abelian groupsMathworldPlanetmath it is always unique, and is called the p-primary component of the abelian group). A maximal p-subgroup may, of course, be trivial. Non-trivial maximal p-subgroups of finite groups are called Sylow p-subgroups (http://planetmath.org/SylowPSubgroups).

Titlep-group
Canonical namePgroup
Date of creation2013-03-22 14:53:08
Last modified on2013-03-22 14:53:08
Owneryark (2760)
Last modified byyark (2760)
Numerical id13
Authoryark (2760)
Entry typeDefinition
Classificationmsc 20F50
Synonymp-groupMathworldPlanetmath
Synonymp-primary group
Synonymprimary group
Related topicPGroup
Related topicPExtension
Related topicProPGroup
Related topicQuasicyclicGroup
Related topicSubgroup
Definesp-subgroup
Definesprimary component
Definesp-primary
Definesp-primary subgroup
Definesprimary subgroup
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更新时间:2025/5/4 23:08:19