Maximum subgroup
In group theory , a subgroup of a given group is called maximal if there is no real subgroup between and . So the subgroup is a maximum subgroup of if applies and there is no really larger subgroup with .
existence
Not all groups have a maximum subgroup. The trivial group trivially has no maximal subgroup. The reviewer group also has no maximum subgroup, because in this group each real subgroup is contained in a larger real subgroup.
properties
If a group has only one maximal subgroup, then it is invariant among all automorphisms , i. H. a characteristic subgroup (and therefore a normal divisor ).
A maximum subgroup is also modular . Because is maximal in and subgroups of with , then is either or (because maximal is). In the first case is . In the second case is .
Maximum subsets are also pronormal .
Frattini group
The intersection of all maximal subgroups of G is called the Frattini group (Frattini subgroup) of G.
Mathematics Subject Classification
In the MSC , examinations of the maximum subgroups are classified under 20E28.
literature
- Vipul Naik: Maximal subgroup ( English ) In: Groupprops . Retrieved May 17, 2014.