mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the term maximal subgroup is used to mean slightly different things in different areas of

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

. 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 In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

, 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 defined dually as an ...

of the

partially ordered set In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...

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'' Group action, acting on a non-empty finite set ''X'' is called primitive if ''G'' acts transitive action, transitively on ''X'' and the only Partition_of_a_set, partitions the ''G''-action preserves are th ...

s of ''G''. They are also much studied for the purposes of

finite group theory In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...

: 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 (just notation, not necessarily the ...

, 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 a 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 In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...

of a unique maximal subgroup.

Existence of maximal subgroup

Any proper subgroup of a finite group is contained in some maximal subgroup, since the proper subgroups form a finite

under inclusion. There are, however, infinite

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...

s that contain no maximal subgroups, for example the Prüfer group.

Maximal normal subgroup

Similarly, a

normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...

''N'' of ''G'' is said to be a maximal normal subgroup (or maximal proper normal subgroup) of ''G'' if ''N'' < ''G'' and there is no normal subgroup ''K'' of ''G'' such that ''N'' < ''K'' < ''G''. We have the following theorem: :Theorem: A normal subgroup ''N'' of a group ''G'' is a maximal normal subgroup if and only if the

quotient In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...

''G''/''N'' is

simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by John ...

Hasse diagrams

These Hasse diagrams show the lattices of subgroups of the

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...

S₄, the

dihedral group In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...

D₄, and C₂³, the third direct power of the

cyclic group In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...

C₂.
The maximal subgroups are linked to the group itself (on top of the Hasse diagram) by an edge of the Hasse diagram.

References

{{DEFAULTSORT:Maximal Subgroup Subgroup properties