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 . ${\displaystyle U}$ ${\displaystyle G}$ ${\displaystyle U}$ ${\displaystyle G}$ ${\displaystyle U}$ ${\displaystyle G}$ ${\displaystyle U\neq G}$ ${\displaystyle H}$ ${\displaystyle U\subsetneq H\subsetneq G}$

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 . ${\displaystyle A}$ ${\displaystyle G}$ ${\displaystyle B,C}$ ${\displaystyle G}$ ${\displaystyle A\leq C}$ ${\displaystyle A=C}$ ${\displaystyle A=G}$ ${\displaystyle A}$ ${\displaystyle \langle A,B\cap C\rangle =A=\langle A,B\rangle \cap C}$ ${\displaystyle \langle A,B\cap C\rangle =\langle A,B\rangle =\langle A,B\rangle \cap C}$

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.