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In mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra.
In group theory, a maximal subgroupH 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 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, 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.[citation needed] 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, the term maximalsubgroup is used to mean slightly different things in different areas of algebra. In group theory, a maximalsubgroup H of a group...
mathematics, a maximal compact subgroup K of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst such...
torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group G is a compact, connected, abelian Lie subgroup of G (and...
elements. A large subgroup H (preferably a maximalsubgroup) of the Monster is selected in which it is easy to perform calculations. The subgroup H chosen is...
The Tits group occurs as a maximalsubgroup of the Fischer group Fi22. The group 2F4(2) also occurs as a maximalsubgroup of the Rudvalis group, as the...
Artinian rings. The ACC is equivalent to the maximal condition: every non-empty collection of subgroups has a maximal member, and the DCC is equivalent to the...
Frattini subgroup Φ ( G ) {\displaystyle \Phi (G)} of a group G is the intersection of all maximalsubgroups of G. For the case that G has no maximal subgroups...
algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general...
· Suz of the Suzuki group. This makes the group 6 · Suz · 2 into a maximalsubgroup of Conway's group Co0 = 2 · Co1 of automorphisms of the Leech lattice...
The maximalsubgroups of Sn fall into three classes: the intransitive, the imprimitive, and the primitive. The intransitive maximalsubgroups are exactly...
{\displaystyle p} , a Sylow p-subgroup (sometimes p-Sylow subgroup) of a group G {\displaystyle G} is a maximal p {\displaystyle p} -subgroup of G {\displaystyle...
Dedekind eta function. Wilson (1999) found the 30 conjugacy classes of maximalsubgroups of B which are listed in the table below. (Gorenstein 1993) Leon,...
mod 3, so is a subgroup of the Chevalley group E8(3). The subgroup preserving the Lie bracket (over the integers) is a maximalsubgroup of the Thompson...
extension of M21 by the symmetric group S3. PΓL(3,4) has an embedding as a maximalsubgroup of M24.(Griess 1998, p. 55) A hyperoval has no 3 points that are collinear...
and only if every maximalsubgroup has prime index. A finite group is supersolvable if and only if every maximal chain of subgroups has the same length...
is a maximalsubgroup of the Lyons group. McL has one conjugacy class of involution (element of order 2), whose centralizer is a maximalsubgroup of type...
G2(4) is in turn isomorphic to a subgroup of the Conway group Co1. There are 9 conjugacy classes of maximalsubgroups of J2. Some are here described in...
stabilizer 3.McL:2. Wilson (1985) found the 9 conjugacy classes of maximalsubgroups of Ly as follows: G2(5) 3.McL:2 53.PSL3(5) 2.A11 51+4:4.S6 35:(2 ×...
contained in a maximalsubgroup of type 211:M24. An image of an octad or 16-set has a centralizer of the form 21+8.O8+(2), a maximalsubgroup. The smallest...
2. The component with det(A) = 1 is SO(n). A maximal torus in a compact Lie group G is a maximalsubgroup among those that are isomorphic to Tk for some...
quasithin groups. Wilson (1984) found the 15 conjugacy classes of maximalsubgroups of Ru as follows: 2F4(2) = 2F4(2)'.2 26.U3(3).2 (22 × Sz(8)):3 23+8:L3(2)...
including character tables, individual conjugacy classes and lists of maximalsubgroup, as well as Schur multipliers and orders of their outer automorphisms...
any subgroup of Co0 that properly contains N; hence N is a maximalsubgroup of Co0 and contains 2-Sylow subgroups of Co0. N also is the subgroup in Co0...
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