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).

Title p-group
Canonical name Pgroup
Date of creation 2013-03-22 14:53:08
Last modified on 2013-03-22 14:53:08
Owner yark (2760)
Last modified by yark (2760)
Numerical id 13
Author yark (2760)
Entry type Definition
Classification msc 20F50
Synonym p-groupMathworldPlanetmath
Synonym p-primary group
Synonym primary group
Related topic PGroup
Related topic PExtension
Related topic ProPGroup
Related topic QuasicyclicGroup
Related topic Subgroup
Defines p-subgroup
Defines primary component
Defines p-primary
Defines p-primary subgroup
Defines primary subgroup