In
mathematics, the term maximal subgroup is used to mean slightly different things in different areas of
algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
.
In
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, 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
In mathematics, especially in order theory, a maximal element of a subset ''S'' of some preordered set is an element of ''S'' that is not smaller than any other element in ''S''. A minimal element of a subset ''S'' of some preordered set is def ...
of the
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) 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 binar ...
of subgroups of ''G'' that are not equal to ''G''. Maximal subgroups are of interest because of their direct connection with
primitive permutation representation
In mathematics, a permutation group ''G'' acting on a non-empty finite set ''X'' is called primitive if ''G'' acts transitively on ''X'' and the only partitions the ''G''-action preserves are the trivial partitions into either a single set or int ...
s 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
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''ยท''y'', or simply ''xy'', ...
, 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