-group
Primary groups
Let be a prime number. A -group (or -primary group) is a group in which the order of every element is a power of . A group that is a -group for some prime is also called a primary group.
Using Lagrange’s Theorem and Cauchy’s Theorem one may show that a finite group is a -group if and only if is a power of .
Primary subgroups
A -subgroup (or -primary subgroup) of a group is a subgroup (http://planetmath.org/Subgroup) of such that is also a -group. A group that is a -subgroup for some prime is also called a primary subgroup.
It follows from Zorn’s Lemma that every group has a maximal -subgroup, for every prime . The maximal -subgroup need not be unique (though for abelian groups it is always unique, and is called the -primary component of the abelian group). A maximal -subgroup may, of course, be trivial. Non-trivial maximal -subgroups of finite groups are called Sylow -subgroups (http://planetmath.org/SylowPSubgroups).
Title | -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-group |
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 |