Maximal subgroup

History
The notion of maximal subgroup probably dates back to the very beginning of group theory.

Symbol-free definition
A maximal subgroup of a group is defined in the following equivalent ways:


 * It is a proper subgroup such that there is no other proper subgroup containing it
 * It is a proper subgroup such that the action of the whole group on its coset space is a primitive group action.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed maximal (in symbols, $$H \le_{\max} G$$ or $$G \ge_{\max} H$$) if it satisfies the following equivalent condition:


 * $$H$$ is a proper subgroup of $$G$$ (i.e. $$H \ne G$$) and if $$H \le K \le G$$ for some subgroup $$K$$, then $$H = K$$ or $$K = G$$.
 * $$H$$ is proper and the action of $$G$$ on the coset space $$G/H$$ is a primitive group action: there is no nontrivial partition of the coset space into blocks such that $$G$$ preserves the partition.

In terms of group actions
In terms of group actions, a subgroup of a group is maximal if the natural group action on its coset space is primitive.

Formalisms
The property of being a maximal subgroup can be expressed in monadic second-order logic: there is no bigger subgroup between the given subgroup and the whole group.

Stronger properties

 * Weaker than::Subgroup of prime index
 * Weaker than::Subgroup of index two
 * Weaker than::Subgroup of double coset index two

Weaker properties

 * Stronger than::NE-subgroup
 * Stronger than::Modular subgroup:
 * Stronger than::Pronormal subgroup:
 * Stronger than::Subgroup contained in finitely many intermediate subgroups

Property bifurcations
There are many pairs of properties such that every maximal subgroup of a group has exactly one of these properties. For a complete list, refer:

Category:Property bifurcations for maximal subgroups

Transfer condition
In general, it may not be true that the intersection of a maximal subgroup with another subgroup is maximal inside that subgroup. If a subgroup has the property that its intersection with every maximal subgroup (not containing it) is maximal in it, the subgroup is termed max-sensitive.

Transiters
The left and right transiters are both the identity element.

Subordination
The subordination property on the property of maximality defines the property of submaximality. For finite groups, every subgroup is submaximal. However, this may not be true in general for infinite groups. It is, however, true that every subgroup of finite index is submaximal.

The maximal operator
The maximal operator is a subgroup property modifier that takes any subgroup property and gives out the property of being a subgroup that is maximal in the group with respect to that property.