PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: Very high Entry average rating: No information on entry rating
maximal subgroup (Definition)

Let $G$ be a group.

A subgroup $H$ of $G$ is said to be a maximal subgroup of $G$ if $H\neq G$ and there is no subgroup $K$ of $G$ such that $H<K<G$. Note that a maximal subgroup of $G$ is not maximal among all subgroups of $G$, but only among all proper subgroups of $G$. For this reason, maximal subgroups are sometimes called maximal proper subgroups.

Similarly, a normal subgroup $N$ of $G$ is said to be a maximal normal subgroup (or maximal proper normal subgroup) of $G$ if $N\neq G$ and there is no normal subgroup $K$ of $G$ such that $N<K<G$. A normal subgroup $N$ of $G$ is a maximal normal subgroup if and only if the quotient $G/N$ is a simple group.



"maximal subgroup" is owned by yark. [ full author list (2) | owner history (1) ]
(view preamble | get metadata)

View style:

See Also: maximal element

Other names:  maximal proper subgroup
Also defines:  maximal, maximal normal subgroup, maximal proper normal subgroup

Attachments:
existence of maximal subgroups (Example) by Algeboy
Log in to rate this entry.
(view current ratings)

Cross-references: simple group, normal subgroup, proper subgroups, subgroup, group
There are 21 references to this entry.

This is version 10 of maximal subgroup, born on 2002-02-19, modified 2007-06-13.
Object id is 2198, canonical name is Maximal.
Accessed 14538 times total.

Classification:
AMS MSC20E28 (Group theory and generalizations :: Structure and classification of infinite or finite groups :: Maximal subgroups)

Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)