Maximal subgroup
Encyclopedia
In mathematics
, the term maximal subgroup is used to mean slightly different things in different areas of algebra
.
In group theory
, a maximal subgroup H of a group
G is a proper subgroup, such that no proper subgroup K contains H strictly. In other words H is a maximal element
of the partially ordered set
of proper subgroups of G. Maximal subgroups are of interest because of their direct connection with primitive permutation representations of G. They are also much studied for the purposes of finite group theory: see for example Frattini subgroup
, the intersection of the maximal subgroups.
In semigroup theory
, a maximal subgroup of a semigroup S is a subgroup (that is, a subsemigroup which forms a group under the semigroup operation) of S which is not properly contained in another subgroup of S. Notice that, here, there is no requirement that the maximal subgroup be proper, so if S is in fact a group then its unique maximal subgroup (as a semigroup) is S itself. Considering subgroups, and in particular maximal subgroups, of semigroups often allows one to apply group-theoretic techniques in semigroup theory. There is a one-to-one correspondence between idempotent elements of a semigroup and maximal subgroups of the semigroup: each idempotent element is the identity element
of a unique maximal subgroup.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the term maximal subgroup is used to mean slightly different things in different areas of algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
.
In group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, a maximal subgroup H of a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
G is a proper subgroup, such that no proper subgroup K contains H strictly. In other words H is a maximal element
Maximal element
In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S. The term minimal element is defined dually...
of the partially ordered set
Partially ordered set
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
of proper subgroups of G. Maximal subgroups are of interest because of their direct connection with primitive permutation representations of G. They are also much studied for the purposes of finite group theory: see for example Frattini subgroup
Frattini subgroup
In mathematics, the Frattini subgroup Φ of a group G is the intersection of all maximal subgroups of G. For the case that G is the trivial group e, which has no maximal subgroups, it is defined by Φ = e...
, the intersection of the maximal subgroups.
In semigroup theory
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
, a maximal subgroup of a semigroup S is a subgroup (that is, a subsemigroup which forms a group under the semigroup operation) of S which is not properly contained in another subgroup of S. Notice that, here, there is no requirement that the maximal subgroup be proper, so if S is in fact a group then its unique maximal subgroup (as a semigroup) is S itself. Considering subgroups, and in particular maximal subgroups, of semigroups often allows one to apply group-theoretic techniques in semigroup theory. There is a one-to-one correspondence between idempotent elements of a semigroup and maximal subgroups of the semigroup: each idempotent element is the identity element
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
of a unique maximal subgroup.
Existence of maximal subgroup
Although every proper subgroup of a finite group is contained in a maximal subgroup, there are infinite Abelian groups that contain no maximal subgroups.Maximal normal subgroup
Similarly, a normal subgroup N of G is said to be a maximal normal subgroup (or maximal proper normal subgroup) of G if N- Theorem: A normal subgroup N of a group G is a maximal normal subgroup if and only if the quotient G/N is simple.