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Term in mathematics
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
Maximal_subgroup
Concept in topology
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
Maximal_compact_subgroup
Sporadic simple group
2 elements. A large subgroup H (preferably a maximal subgroup) of the Monster is selected in which it is easy to perform calculations. The subgroup H chosen is
Monster_group
Sporadic simple group
extension of M21 by the symmetric group S3. PΓL(3,4) has an embedding as a maximal subgroup of M24.(Griess 1998, p. 55) A hyperoval has no 3 points that are collinear
Mathieu_group_M24
Theorems that help decompose a finite group based on prime factors of its order
p} . A Sylow p-subgroup (sometimes p-Sylow subgroup) of a finite group G {\displaystyle G} is a maximal p {\displaystyle p} -subgroup of G {\displaystyle
Sylow_theorems
Type of group in abstract algebra
The maximal subgroups of Sn fall into three classes: the intransitive, the imprimitive, and the primitive. The intransitive maximal subgroups are exactly
Symmetric_group
Finite simple group; sometimes classed as sporadic
The Tits group occurs as a maximal subgroup of the Fischer group Fi22. The group 2F4(2) also occurs as a maximal subgroup of the Rudvalis group, as the
Tits_group
Maximal compact connected Abelian Lie subgroup
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
Maximal_torus
Sporadic simple group
is a maximal subgroup of the Lyons group. McL has one conjugacy class of involution (element of order 2), whose centralizer is a maximal subgroup of type
McLaughlin_sporadic_group
Sporadic simple group
Dedekind eta function. Wilson (1999) found the 30 conjugacy classes of maximal subgroups of B which are listed in the table below. (Gorenstein 1993) Leon,
Baby_monster_group
Intersection of all maximal subgroups
Frattini subgroup Φ ( G ) {\displaystyle \Phi (G)} of a group G is the intersection of all maximal subgroups of G. For the case that G has no maximal subgroups
Frattini_subgroup
Sporadic simple group
computer-free construction of the Lyons group, as an amalgam of its maximal 3-local subgroups. When the McLaughlin sporadic group was discovered, it was noticed
Lyons_group
Sporadic simple group
found the 13 conjugacy classes of maximal subgroups of J4 which are listed in the table below. A Sylow 3-subgroup of J4 is a Heisenberg group: order
Janko_group_J4
Sporadic simple group
{\displaystyle \mathbb {Z} /2\mathbb {Z} \times \mathrm {Co} _{3}} . Some maximal subgroups fix or reflect 2-dimensional sublattices of the Leech lattice. It
Conway_group_Co3
Sporadic simple group
and Yoshiara (1985) independently found the 13 conjugacy classes of maximal subgroups of O'N as follows: In 2017 John F. R. Duncan, Michael H. Mertens,
O'Nan_group
248-dimensional exceptional simple Lie group
Lie group of real dimension 496. This is simply connected, has maximal compact subgroup the compact form (see below) of E8, and has an outer automorphism
E8_(mathematics)
Classification theorem in group theory
elements of a maximal abelian subgroup. The normalizers of these maximal abelian subgroups turn out to be exactly the maximal proper subgroups of G. These
Feit–Thompson_theorem
Sporadic simple group
found the 11 conjugacy classes of maximal subgroups of He as follows: Butler, Gregory (1981), "The maximal subgroups of the sporadic simple group of Held"
Held_group
Sporadic simple group
A003918 in the OEIS). Wilson (1984) found the 15 conjugacy classes of maximal subgroups of Ru as follows: Griess (1982) Aschbacher, Michael; Smith, Stephen
Rudvalis_group
Sporadic simple group
mod 3, so is a subgroup of the Chevalley group E8(3). The subgroup preserving the Lie bracket (over the integers) is a maximal subgroup of the Thompson
Thompson_sporadic_group
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
Subgroup_series
Sporadic simple group
contained in a maximal subgroup of type 211:M24. An image of an octad or 16-set has a centralizer of the form 21+8.O+ 8(2), a maximal subgroup. The smallest
Conway_group_Co1
Sporadic simple group
function. Norton & Wilson (1986) found the 14 conjugacy classes of maximal subgroups of HN as follows: Harada, Koichiro (1976), "On the simple group F
Harada–Norton_group
Algebraic structure
element of the subgroup. For each idempotent e of the semigroup there is a unique maximal subgroup containing e. Each maximal subgroup arises in this
Semigroup
Natural number
largest prime factor. M 11 {\displaystyle \mathrm {M} _{11}} is the maximal subgroup Mathieu group M 12 {\displaystyle \mathrm {M} _{12}} , where 11 is
11_(number)
Type of subgroup of an algebraic group
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
Borel_subgroup
Sporadic simple group
of the monster vertex algebra. There are 11 conjugacy classes of maximal subgroups of M12, 6 occurring in automorphic pairs, as follows: The cycle shape
Mathieu_group_M12
Sporadic simple group
group J4. There are no proper subgroups transitive on all 22 points. There are 8 conjugacy classes of maximal subgroups of M22 as follows: There are 12
Mathieu_group_M22
Maximal connected Abelian subgroup
closed) field k {\displaystyle k} is the centralizer of a maximal torus. Cartan subgroups are smooth (equivalently reduced), connected and nilpotent
Cartan_subgroup
Sporadic simple group
related to the double cover of the subgroup M22. Magliveras (1971) found the 12 conjugacy classes of maximal subgroups of HS as follows: Traces of matrices
Higman–Sims_group
Sporadic simple group
representations of the Mathieu group M24. There are 7 conjugacy classes of maximal subgroups of M23 as follows: Cameron, Peter J. (1999), Permutation Groups, London
Mathieu_group_M23
Sporadic simple group
} Finkelstein & Rudvalis (1974) found the 9 conjugacy classes of maximal subgroups of J3 as follows: Griess (1982): p. 93: proof that J3 is a pariah
Janko_group_J3
Sporadic simple group
representations of M11 over any field. There are 5 conjugacy classes of maximal subgroups of M11 as follows: The maximum order of any element in M11 is 11.
Mathieu_group_M11
Sporadic simple group
found the 7 conjugacy classes of maximal subgroups of J 1 {\displaystyle J_{1}} shown in the table. Maximal simple subgroups of order 660 afford J 1 {\displaystyle
Janko_group_J1
Sporadic simple group
G2(4) is in turn isomorphic to a subgroup of the Conway group Co1. There are 9 conjugacy classes of maximal subgroups of J2. Some are here described in
Janko_group_J2
Fitting subgroup, so the generalized Fitting subgroup is a central extension of a product of p-groups and simple groups. The layer is also the maximal normal
Fitting_subgroup
Four finite groups derived from the Leech lattice
any subgroup of Co0 that properly contains N; hence N is a maximal subgroup of Co0 and contains 2-Sylow subgroups of Co0. N also is the subgroup in Co0
Conway_group
Sporadic simple group
· Suz of the Suzuki group. This makes the group 6 · Suz · 2 into a maximal subgroup of Conway's group Co0 = 2 · Co1 of automorphisms of the Leech lattice
Suzuki_sporadic_group
Theorem in group theory
groups is what makes it so useful. Originally the theorem was about maximal subgroups of the symmetric group. It appeared as an appendix to a paper by Leonard
O'Nan–Scott_theorem
theory describes the closed connected subgroups of a compact Lie group that have maximal rank, i.e. contain a maximal torus. It is named after the Swiss
Borel–de_Siebenthal_theory
Finite simple group type not classified as Lie, cyclic or alternating
including character tables, individual conjugacy classes and lists of maximal subgroup, as well as Schur multipliers and orders of their outer automorphisms
Sporadic_group
the subgroup generated by the abelian subgroups of P of maximal order or the subgroup generated by the elementary abelian subgroups of P of maximal rank
Thompson_subgroup
Five sporadic simple groups
constructed in various ways. M12 has a simple subgroup of order 660, a maximal subgroup. That subgroup is isomorphic to the projective special linear
Mathieu_group
Mathematical group
the Borel subgroups are also defined to be the maximal closed and connected solvable subgroups, and the parabolic subgroups are the subgroups that contain
Parabolic subgroup of a reflection group
Parabolic_subgroup_of_a_reflection_group
Sporadic simple group
vector of type 2. It is thus a subgroup of Co0. It is isomorphic to a subgroup of Co1. The direct product 2×Co2 is maximal in Co0. The Schur multiplier
Conway_group_Co2
Special group in linear algebra
conjugacy classes of maximal parahorics. When G is commutative, it has a unique maximal compact subgroup and a unique Iwahori subgroup, which is contained
Iwahori_subgroup
Type of mathematical group
many maximal Kleinian groups. A principal congruence subgroup of Γ = S L 2 ( Z ) {\displaystyle \Gamma =\mathrm {SL} _{2}(\mathbb {Z} )} is a subgroup of
Arithmetic_Fuchsian_group
Mathematical term in group theory
is closed in the pro-finite topology on G {\displaystyle G} . Every maximal subgroup of G {\displaystyle G} has finite index in G {\displaystyle G} . The
Grigorchuk_group
Operator in quantum field theory
the generator of the little group of the Poincaré group, that is the maximal subgroup (with four generators) leaving the eigenvalues of the four-momentum
Pauli–Lubanski_pseudovector
Type of group in mathematics
2. The component with det(A) = 1 is SO(n). A maximal torus in a compact Lie group G is a maximal subgroup among those that are isomorphic to Tk for some
Orthogonal_group
Natural number
holds a total of 193 conjugacy classes. It also holds at least 44 maximal subgroups aside from the double cover of B {\displaystyle \mathbb {B} } (the
193_(number)
Sporadic simple group
06.020, ISSN 0021-8693, MR 2303203 Wilson, Robert A. (1984), "On maximal subgroups of the Fischer group Fi22", Mathematical Proceedings of the Cambridge
Fischer_group_Fi22
Group with series of normal subgroups where all factors are cyclic
and only if every maximal subgroup has prime index. A finite group is supersolvable if and only if every maximal chain of subgroups has the same length
Supersolvable_group
Finite group
sporadic group (the full automorphism group of this lattice) as a maximal subgroup. Huppert (1967, p.124) showed that any extension of G L n ( F q ) {\displaystyle
Dempwolff_group
Sporadic simple group
\end{aligned}}} Linton & Wilson (1991) found the 25 conjugacy classes of maximal subgroups of Fi24' as follows: Aschbacher, Michael (1997), 3-transposition groups
Fischer_group_Fi24
precisely the subgroup(Curtis & Reiner 1981, p.354). Equivalently, an SA subgroup is a centrally closed abelian subgroup. Any SA subgroup is a maximal abelian
Special_abelian_subgroup
Sporadic simple group
Kleidman, Parker & Wilson (1989) found the 14 conjugacy classes of maximal subgroups of Fi23 as follows: Aschbacher, Michael (1997), 3-transposition groups
Fischer_group_Fi23
78-dimensional exceptional simple Lie group
has maximal compact subgroup SO(2) × Spin(10)/(center), fundamental group Z and trivial outer automorphism group. EIV (or E6(-26)), which has maximal compact
E6_(mathematics)
Simple Lie group; the automorphism group of the octonions
conjugation as an outer automorphism and is simply connected. The maximal compact subgroup of its associated group is the compact form of G2. The Lie algebra
G2_(mathematics)
Nilpotent subalgebra of a Lie algebra
‘Cartan subgroup,’ especially when dealing with disconnected groups. For compact connected Lie groups, a Cartan subgroup is essentially a maximal connected
Cartan_subalgebra
combinatorial in this case, respected by translation shows that either K is a maximal subgroup of G, or there is a system of imprimitivity (roughly, a lack of full
System_of_imprimitivity
133-dimensional exceptional simple Lie group
group of real dimension 266. This has fundamental group Z/2Z, has maximal compact subgroup the compact form (see below) of E7, and has an outer automorphism
E7_(mathematics)
On certain subgroups of a minimal simple finite group of odd order
finite group of odd order there is a unique maximal subgroup containing a given elementary abelian subgroup of rank 3. Bender (1970) gave a shorter proof
Thompson_uniqueness_theorem
Group theory method by Bender
method involves studying a maximal subgroup M containing the centralizer of an involution, and its generalized Fitting subgroup F*(M). One succinct version
Bender's_method
Nilpotent, self-normalizing subgroup
order six is a maximal nilpotent subgroup, but only those of order two are Carter subgroups. Every subgroup containing a Carter subgroup of a soluble group
Carter_subgroup
British mathematician
Bray and Derek Holt, Roney-Dougal is the co-author of the book The Maximal Subgroups of the Low-Dimensional Finite Classical Groups (London Mathematical
Colva_Roney-Dougal
Mathematician (born 1958)
London, England, who is best known for his work on classifying the maximal subgroups of finite simple groups and for the work in the Monster group. He
Robert_Arnott_Wilson
Mathematical term in group theory
subgroups. As there is no maximal subgroup of a Prüfer p-group, it is its own Frattini subgroup. Given this list of subgroups, it is clear that the Prüfer
Prüfer_group
Natural number
that is not the sum of two abundant numbers), which is the number of maximal subgroups of the friendly giant F 1 {\displaystyle \mathbb {F} _{1}} , the largest
92_(number)
Automorphism group of the Klein quartic
imaginary quadratic number fields of class number 1. PSL(2, 7) is a maximal subgroup of the Mathieu group M21; the groups M21 and M24 can be constructed
PSL(2,7)
statement that every maximal conjugate permutable subgroup is normal. (The finiteness is used crucially in the proofs.) In summary, a subgroup H of a finite
Quasinormal_subgroup
Monster and modular connection
Providence: Amer. Math. Soc. pp. 303–313. Atlas of finite groups: maximal subgroups and ordinary characters for simple groups. John H. Conway. Oxford
Monstrous_moonshine
finite group: Every maximal conjugate-permutable subgroup is normal. Every conjugate-permutable subgroup is a conjugate-permutable subgroup of every intermediate
Conjugate-permutable_subgroup
following is a straight-line program that computes a generating set for a maximal subgroup E32·E32⋊C31. This straight-line program can be found in the online
Straight-line_program
S. P.; Parker, R. A.; and Wilson, R. A.: "Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups." Oxford, England 1985.
List_of_finite_simple_groups
Permutation group that preserves no non-trivial partition
the set G/H of cosets for H a subgroup of G. A group action is primitive if it is isomorphic to G/H for a maximal subgroup H of G, and imprimitive otherwise
Primitive_permutation_group
Dynkin diagram of a real semisimple Lie algebra that indicates the maximal compact subgroup. Although they resemble Satake diagrams they are a different way
Vogan_diagram
of these is a (maximal) subgroup. In particular, the third D-class is isomorphic to the symmetric group S3. There are also six subgroups of order 2, and
Green's_relations
Invariant of polynomial roots
One way is to begin from maximal (transitive) subgroups until the right one is found and then continue with maximal subgroups of that. "Resultants, Resolvents
Resolvent_(Galois_theory)
7-regular undirected graph with 50 nodes and 175 edges
alternating group on 6 letters. Both of the two types of stabilizers are maximal subgroups of the whole automorphism group of the Hoffman–Singleton graph. The
Hoffman–Singleton_graph
Theorem classifying finite simple groups
involution, C/O(C) has a component (where O(C) is the core of C, the maximal normal subgroup of odd order). These are more or less the groups of Lie type of
Classification of finite simple groups
Classification_of_finite_simple_groups
Aspect of mathematical group theory
automorphism group of A6 appears naturally as a maximal subgroup of the Mathieu group M12 in 2 ways, as either a subgroup fixing a division of the 12 points into
Automorphisms of the symmetric and alternating groups
Automorphisms_of_the_symmetric_and_alternating_groups
p-group S: the subgroup generated by the abelian subgroups of maximal order. Z(H) means the center of a group H. Op′ is the maximal normal subgroup of G of order
ZJ_theorem
{x}) or the stabilizer Gx is a maximal subgroup of G (then the stabilizers of all elements of X are the maximal subgroups of G conjugate to Gx because Ggx
Block (permutation group theory)
Block_(permutation_group_theory)
Theory of strings with supersymmetry
thought, however, that 16 is probably the maximum since SO(16) is a maximal subgroup of E8, the largest exceptional Lie group, and also is more than large
Superstring_theory
Theorem in algebra
context-free word problem as being precisely those with a virtually free maximal subgroup. Subsequent to the 1983 paper of Muller and Schupp, several authors
Muller–Schupp_theorem
Topics referred to by the same term
Vogan, Togo, a town Vogan diagram, a diagram that indicates the maximal compact subgroup Vågan This disambiguation page lists articles associated with the
Vogan
Subgroup of the group of invertible n×n matrices
field k, a Borel subgroup of G means a maximal smooth connected solvable subgroup. For example, one Borel subgroup of GL(n) is the subgroup B of upper-triangular
Linear_algebraic_group
Open convex self-dual cones
are maximal compact subgroups, all conjugate, and exhaust the maximal compact subgroups of Aut C. In Aut0 C the stabilizers of points are maximal compact
Symmetric_cone
From an exceptional automorphism of a Dynkin diagram
Janko found the sporadic group J1. Kleidman (1988) determined their maximal subgroups. The Ree groups of type 2G2 are exceptionally hard to characterize
Ree_group
Frattini subgroup of a K-group is trivial; if a group has a core-free maximal subgroup that is a K-group, then it itself is a K-group; hence subgroups of K-groups
Complemented_group
case, the Frattini subgroup, which is defined as the intersection of all maximal subgroups coincides with the commutator subgroup. Proof. To see this
Artin_transfer_(group_theory)
under convolution. It can also be defined for a pair (g, K) of a maximal compact subgroup K of a Lie group with Lie algebra g, in which case the Hecke algebra
Hecke_algebra_of_a_pair
avoiding computer calculations. Similarly, the calculation of the maximal subgroups of the larger sporadic groups uses a lot of computer calculations
List of long mathematical proofs
List_of_long_mathematical_proofs
are 36 simple maximal subgroups of order 168. These are the vertices of a subgraph, the U3(3) graph. A 168-subgroup has 14 maximal subgroups of order 24
Hall–Janko_graph
Infinite family of simple groups of Lie type
has at least 4 types of maximal subgroups. The diagonal subgroup is cyclic, of order q – 1. The lower triangular (Borel) subgroup and its conjugates, of
Suzuki_groups
Subgroup of a root system's isometry group
always be realized as a subgroup of G. If B is a Borel subgroup of G, i.e., a maximal connected solvable subgroup and a maximal torus T = T0 is chosen
Weyl_group
P.; Parker, R. A.; Wilson, R. A. (1985). Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. with computational assistance
Higman–Sims_graph
Mathematics book by John Conway
various constructions (such as presentations), conjugacy classes of maximal subgroups, and, most importantly, character tables (including power maps on
ATLAS_of_Finite_Groups
In mathematics, a minimal K-type is a representation of a maximal compact subgroup K of a semisimple Lie group G that is in some sense the smallest representation
Minimal_K-type
MAXIMAL SUBGROUP
MAXIMAL SUBGROUP
Male
French
French form of Latin Maximus, MAXIME means "the greatest."Â
Boy/Male
Muslim
Liberal, Generous, Another name for God
Boy/Male
Hindu, Indian
Praise
Boy/Male
Sikh
A king
Boy/Male
Hindu, Indian, Marathi
The Garland of Lord Vishnu
Boy/Male
Hindu, Indian, Punjabi, Sikh, Tamil
Great Speech
Girl/Female
Muslim
Soft
Boy/Male
Celebrity, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Punjabi, Sanskrit, Sikh, Tamil, Telugu, Traditional
King of the Earth; A King
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Sindhi, Tamil, Telugu, Traditional
A String of Pearls
Girl/Female
Hindu
Greatness
Boy/Male
British, English
Ermine; Ferret-like Mammal; Animal Name
Male
Russian
(МакÑим) Variant spelling of Russian Maksim, MAXIM means "the greatest." Compare with another form of Maxim.
Girl/Female
Indian
Soft
Boy/Male
American, Australian, Chinese, French, German, Greek, Latin, Swedish
Greatest
Boy/Male
Hindu
Dignity, Power
Boy/Male
Gujarati, Hindu, Indian
Rich; Maladar
Boy/Male
Hindu
Devoted, A promise to God
Girl/Female
Hindu
Full of jewel
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Fragrance
Boy/Male
Latin
Greatest.
MAXIMAL SUBGROUP
MAXIMAL SUBGROUP
Male
Polish
Polish form of Latin Nicodemus, NIKODEM means "victory of the people."
Girl/Female
Native American
Spirit.
Girl/Female
Australian, German
Guardian
Boy/Male
Latin
Prisoner.
Boy/Male
Indian, Sanskrit
Injurer
Girl/Female
Indian, Tamil
Bird
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Sindhi, Tamil, Telugu
Sage Katyanan Worships this Name; Goddess Parvati / Durga; Goddess Parvati
Boy/Male
American, British, English, French, Latin
Bean Grower; Derived from the Roman Clan Name Fabius; A Name Given Several Roman Emperors and 16 Saints; One who Grows Beans
Boy/Male
Arabic, Indian, Muslim
Masters; Lords
Surname or Lastname
English (of Norman origin)
English (of Norman origin) : possibly a habitational name from Chenevray in Haute-Saône, France.
MAXIMAL SUBGROUP
MAXIMAL SUBGROUP
MAXIMAL SUBGROUP
MAXIMAL SUBGROUP
MAXIMAL SUBGROUP
n.
One of the lower or outer jaws of arthropods.
a.
Partaking of the nature both of vegetable and animal matter; -- a term sometimes applied to vegetable albumen and gluten, from their resemblance to similar animal products.
v.
Of or pertaining to a husband; as, marital rights, duties, authority.
n.
The greatest quantity or value attainable in a given case; or, the greatest value attained by a quantity which first increases and then begins to decrease; the highest point or degree; -- opposed to minimum.
n.
The bone, or principal bone, of the upper jaw, the bone of the lower jaw being the mandible.
n.
One of the Mammalia.
a.
Relating to the morning, or to matins; matutinal.
mexcal.
See Mescal.
pl.
of Maximum
n.
One of the lower animals; a brute or beast, as distinguished from man; as, men and animals.
a.
Consisting of the flesh of animals; as, animal food.
a.
Pertaining to the merely sentient part of a creature, as distinguished from the intellectual, rational, or spiritual part; as, the animal passions or appetites.
n.
An organized living being endowed with sensation and the power of voluntary motion, and also characterized by taking its food into an internal cavity or stomach for digestion; by giving carbonic acid to the air and taking oxygen in the process of respiration; and by increasing in motive power or active aggressive force with progress to maturity.
a.
Performed by, or proceeding from, occult and superhuman agencies; done by, or seemingly done by, enchantment or sorcery. Hence: Seemingly requiring more than human power; imposing or startling in performance; producing effects which seem supernatural or very extraordinary; having extraordinary properties; as, a magic lantern; a magic square or circle.
n.
An animal that suckles its young; a mammal.
a.
Greatest in quantity or highest in degree attainable or attained; as, a maximum consumption of fuel; maximum pressure; maximum heat.
a.
Belonging to the axis of the body; as, the axial skeleton; or to the axis of any appendage or organ; as, the axial bones.
a.
Of or relating to animals; as, animal functions.
a.
Pertaining to the hidden wisdom supposed to be possessed by the Magi; relating to the occult powers of nature, and the producing of effects by their agency.
n.
The bone of either the upper or the under jaw.