maximal subgroup
Let be a group.
A subgroup of
is said to be a maximal subgroup of
if and there is no subgroup of
such that .
Note that a maximal subgroup of is not maximal (http://planetmath.org/MaximalElement) among all subgroups of ,
but only among all proper subgroups
of .
For this reason, maximal subgroups are sometimes called maximal proper subgroups.
Similarly, a normal subgroup of
is said to be a maximal normal subgroup
(or maximal proper normal subgroup) of
if and there is no normal subgroup of
such that .
We have the following theorem:
Theorem.
A normal subgroup of a group is a maximal normal subgroup if and only if the quotient (http://planetmath.org/QuotientGroup) is simple (http://planetmath.org/Simple).
Title | maximal subgroup |
Canonical name | MaximalSubgroup |
Date of creation | 2013-03-22 12:23:46 |
Last modified on | 2013-03-22 12:23:46 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 15 |
Author | yark (2760) |
Entry type | Definition |
Classification | msc 20E28 |
Synonym | maximal proper subgroup |
Related topic | MaximalElement |
Defines | maximal |
Defines | maximal normal subgroup |
Defines | maximal proper normal subgroup |
Defines | simplicity of quotient group |