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

algebra Algebra () is one of the 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 mathematics. Elementary a ...

. In

group theory In 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 ...

, a maximal subgroup ''H'' of a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

''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 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 binary ...

of subgroups of ''G'' that are not equal to ''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 In mathematics, particularly in group theory, the Frattini subgroup \Phi(G) of a group is the intersection of all maximal subgroups of . For the case that has no maximal subgroups, for example the trivial group or a Prüfer group, it is def ...

, 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 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 operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...

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 comm ...

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

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'' < ''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 lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...

''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 Johnn ...

Hasse diagrams

These

Hasse diagram In order theory, a Hasse diagram (; ) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set ''(S, ≤)'' one represents e ...

s show the lattices of subgroups of the symmetric group S₄, the dihedral group D₄, and C₂³, the third direct power of the cyclic group 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