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the Thompson subgroup J ( P ) {\displaystyle J(P)} of a finite p-group P refers to one of several characteristic subgroups of P. John G. Thompson (1964)
Thompson_subgroup
Topics referred to by the same term
Thompson group or Thompson's group can refer to either The finite Thompson sporadic group Th studied by John G. Thompson The finite Thompson subgroup
Thompson_group
Characteristic subgroup in mathematical finite group theory
the Puig subgroup, introduced by Puig (1976), is a characteristic subgroup of a p-group analogous to the Thompson subgroup. If H is a subgroup of a group
Puig_subgroup
American mathematician
Letters. Feit–Thompson theorem McKay–Thompson series Quadratic pair Thompson factorization Thompson order formula Thompson subgroup Thompson transitivity
John_G._Thompson
Finite group
using all non-trivial subgroups of a Sylow p-subgroup, one uses only the non-trivial characteristic subgroups. For odd primes p Thompson found such a strengthened
Normal_p-complement
Sporadic simple group
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 group called
Thompson_sporadic_group
Classification theorem in group theory
maximal subgroups of G (up to conjugacy) is repeated in both the proof of the Feit–Hall–Thompson CN-theorem and in the proof of the Feit–Thompson odd-order
Feit–Thompson_theorem
Three groups
The group F is not simple but its derived subgroup [F,F] is and the quotient of F by its derived subgroup is the free abelian group of rank 2. F is totally
Thompson_groups
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
normal p-subgroup for some odd prime p, then Op′(G)Z(J(S)) is a normal subgroup of G, for any Sylow p-subgroup S. J(S) is the Thompson subgroup of a p-group
ZJ_theorem
Interior Salishan language
name of the subgroup of the Nlakaʼpamux who live there. Thompson is a consonant-heavy language. The consonants can be divided into two subgroups: obstruents
Thompson_language
Matrix group
subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example is the subgroup of
Congruence_subgroup
Theorem classifying finite simple groups
odd order, which are all solvable by the Feit–Thompson theorem. Groups of 2-rank 1. The Sylow 2-subgroups are either cyclic, which is easy to handle using
Classification of finite simple groups
Classification_of_finite_simple_groups
mathematical group theory, the Thompson replacement theorem is a theorem about the existence of certain abelian subgroups of a p-group. The Glauberman replacement
Replacement_theorem
Sporadic simple group
extension of O by A8. (Thompson 1983, pp. 197–208) A duum is a pair of complementary dodecads (12 point sets) in the Golay code. The subgroup fixing a duad is
Mathieu_group_M24
Mathematical theory
mathematics, a Thompson factorization, introduced by Thompson (1966), is an expression of some finite groups as a product of two subgroups, usually normalizers
Thompson_factorization
the Thompson transitivity theorem gives conditions under which the centralizer of an abelian subgroup A acts transitively on certain subgroups normalized
Thompson_transitivity_theorem
Concept in mathematics
In mathematics, or more specifically group theory, the omega and agemo subgroups described the so-called "power structure" of a finite p-group. They were
Omega_and_agemo_subgroup
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 of
Thompson_uniqueness_theorem
Mathematical group
are simple if n is even and have a simple subgroup of index 2 if n is odd, one of which is one of the Thompson groups. Higman's group is generated by 4
Higman_group
Four finite groups derived from the Leech lattice
square norm. Subgroups are often named in reference to the types of relevant fixed points. This lattice has no vectors of type 1. Thomas Thompson (1983) relates
Conway_group
Any of certain special normal subgroups of a group
is any of certain special normal subgroups of a group. The two most common types are the normal core of a subgroup and the p-core of a group. For a group
Core_(group_theory)
whose local subgroups (that is, the normalizers of nontrivial p-subgroups) are solvable groups. The non-solvable ones were classified by Thompson during his
N-group_(finite_group_theory)
of finite simple groups Alternating group Borel subgroup Chevalley group Conway group Feit–Thompson theorem Fischer group General linear group Group
List_of_group_theory_topics
Sporadic simple group
eta function. Wilson (1999) found the 30 conjugacy classes of maximal subgroups of B which are listed in the table below. (Gorenstein 1993) Leon, Jeffrey
Baby_monster_group
Sporadic simple group
sporadic groups and was discovered by Jack McLaughlin (1969) as an index 2 subgroup of a rank 3 permutation group acting on the McLaughlin graph with 275 =
McLaughlin_sporadic_group
Sporadic simple group
1986 Robert A. Wilson showed that J 1 {\displaystyle J_{1}} cannot be a subgroup of the monster group. Thus it is one of the 6 sporadic groups called the
Janko_group_J1
Group with subnormal series where all factors are abelian
solvable group is a group whose derived series terminates in the trivial subgroup. Historically, the word "solvable" arose from Galois theory and the proof
Solvable_group
years, as Amish churches have divided many times over doctrinal disputes, subgroups have developed. The "Old Order Amish", a conservative faction that withdrew
Subgroups_of_Amish
Algebraic structure
1965) in order to extend Thompson's uniqueness results in the odd order theorem to groups with dihedral Sylow 2-subgroups. There are several equivalent
P-stable_group
Monster and modular connection
Thompson studied the quotient of the hyperbolic plane by subgroups of SL2(R), particularly, the normalizer Γ0(p)+ of the Hecke congruence subgroup Γ0(p)
Monstrous_moonshine
Concept in mathematics
has the property that every subgroup whose order is the product of 2 primes is cyclic; this implies that its Sylow subgroups are cyclic or generalized quaternion
Frobenius_group
Sporadic simple group
subgroup of the Monster group, the full centralizer of a transposition is the double cover of the Baby monster group. As a result, Fi23 is a subgroup
Fischer_group_Fi23
Theorem in group theory
{\displaystyle G} is a finite group, and N {\displaystyle N} is a normal subgroup whose order is coprime to the order of the quotient group G / N {\displaystyle
Schur–Zassenhaus_theorem
type is a p-group such that all characteristic abelian subgroups are cyclic. According to Thompson (1968, p.386), the p-groups of symplectic type were classified
Group_of_symplectic_type
Group without normal subgroups other than the trivial group and itself
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple
Simple_group
3D symmetry group
orientation-preserving symmetries forms a group referred to as the alternating subgroup A4 of S4. Chiral and full (or achiral tetrahedral symmetry and pyritohedral
Tetrahedral_symmetry
Sporadic simple group
3-cycles is normalized by the Fischer group Fi24, so He:2 is a subgroup of the derived subgroup Fi24' (the non-simple group Fi24 has 2 conjugacy classes of
Held_group
area of algebra known as group theory, a complement of a subgroup H in a group G is a subgroup K of G such that G = H K = { h k : h ∈ H , k ∈ K } and
Complement_(group_theory)
Finite simple group; sometimes classed as sporadic
by Jacques Tits (1964) who showed that it is almost simple, its derived subgroup 2F4(2)′ of index 2 being a new simple group, now called the Tits group
Tits_group
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
Sporadic simple group
centralized by the Baby monster group, which therefore contains HN as a subgroup. Conway and Norton suggested in their 1979 paper that monstrous moonshine
Harada–Norton_group
Sporadic simple group
construction of the Lyons group, as an amalgam of its maximal 3-local subgroups. When the McLaughlin sporadic group was discovered, it was noticed that
Lyons_group
Finite simple group type not classified as Lie, cyclic or alternating
is a subgroup S4×2F4(2)′ normalizing a 2C2 subgroup of the Baby monster B, giving rise to a subgroup 2·S4×2F4(2)′ normalizing a certain Q8 subgroup of the
Sporadic_group
Finite group
E_{8}} as the subgroup fixing a certain lattice in the Lie algebra of E 8 {\displaystyle E_{8}} , and is also contained in the Thompson sporadic group
Dempwolff_group
only stated that the corresponding groups have a similar subgroup structure as Ree groups. Thompson (1967, 1972, 1977) and Bombieri, Odlyzko & Hunt (1980)
Walter_theorem
3D symmetry group
direct product S4 × S2, one can simply identify the elements of tetrahedral subgroup Td as a ∈ [ 0 , 4 ! ) {\displaystyle a\in [0,4!)} and their inversions
Octahedral_symmetry
Street gang operating on the East Coast and the South in the United States
(NYCHA) development. Sex, Money, Murder is one of the original sets (subgroups) of the United Blood Nation. Sex, Money, Murder is a Bronx-based street
Sex_Money_Murder
instance, a non-solvable CN-group G is such that its largest solvable normal subgroup O∞(G) is a 2-group, and the quotient is a group of even order. Solvable
CN-group
Mathematical group based upon a finite number of elements
classification of finite simple groups (those with no nontrivial normal subgroup) was completed in 2004. During the twentieth century, mathematicians investigated
Finite_group
Set with associative invertible operation
notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties
Group_(mathematics)
held the titles as a Bullet Club contingent under the House of Torture subgroup until May 3, at Wrestling Dontaku where the latter split up from Bullet
2025 in professional wrestling
2025_in_professional_wrestling
Election of Joe Biden
original on June 15, 2018. Retrieved June 15, 2018. Korecki, Natasha; Thompson, Alex (March 11, 2019). "DNC picks Milwaukee to host 2020 convention".
2020 United States presidential election
2020_United_States_presidential_election
They are part of the larger Nlaka'pamux (Thompson) people and closely connected to the Scw'exmx, a major subgroup of the Nlakapamuxtsin-speaking peoples
Sxeʼxnʼx
Disproven mathematical theory concerning Banach-Tarski and amenable groups
conjecture stated that a group G is non-amenable if and only if G contains a subgroup that is a free group on two generators. The conjecture was disproved in
Von_Neumann_conjecture
U.S Armed conflict in South Asia
the original on 23 June 2021. Retrieved 6 October 2017. Ghosh, Bobby; Thompson, Mark (1 June 2009). "The CIA's Silent War in Pakistan". Time. Retrieved
War in Afghanistan (2001–2021)
War_in_Afghanistan_(2001–2021)
David; Dobrzanski, Edward; Robson, Sean; Cuggy, Michael; Demski, Matthew; Thompson, Deborah (2012). "Great Canadian Lagerstätten 3. Late Ordovician Konservat-Lagerstätten
List_of_lagerstätten
conjugate to z, and x in the subgroup generated by uv. Harris (1972, 3.10) gives the following more complicated version of the Thompson order formula for the
Thompson_order_formula
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 faithful
Conway_group_Co1
Group in which the order of every element is a power of p
Given a finite group G, the Sylow theorems guarantee the existence of a subgroup of G of order pn for every prime power pn that divides the order of G.
P-group
Second election of Donald Trump
Harris and Trump are visiting most". Axios. Retrieved August 1, 2025. Thompson, Alex (August 1, 2024). "Harris ditches Biden's strategy with "freedom"
2024 United States presidential election
2024_United_States_presidential_election
Sporadic simple group
group of units has order 211 − 1 = 2047 = 23 · 89, so it has a cyclic subgroup C of order 23. The Mathieu group M23 can be identified with the group of
Mathieu_group_M23
& Thompson (1963, p.780) defined a three-step group to be a group G satisfying the following conditions: The derived group of G is a Hall subgroup with
3-step_group
English mathematician (1904–1982)
Isoclinism of groups Regular p-group Three subgroups lemma Hall algebra, and Hall polynomials Hall subgroup Hall–Higman theorem Hall–Littlewood polynomial
Philip_Hall
Group of indigenous languages of North America
in its own tentative subgroup. Tsetsaut subgroup Tsetsaut (also known as Tsʼetsʼaut, Wetalh) Central British Columbia subgroup (also known as "British
Athabaskan_languages
Sporadic simple group
been implicitly found earlier by Coxeter (1958), who showed that M12 is a subgroup of the projective linear group of dimension 6 over the finite field with
Mathieu_group_M12
the commutator subgroup of [4,3,3]. A high-index reflective subgroup is the prismatic octahedral symmetry, [4,3,2] (), order 96, subgroup index 4, (Du Val
Point groups in four dimensions
Point_groups_in_four_dimensions
Group of geometric symmetries with at least one fixed point
taken to be a fixed point, and every point group in dimension d is then a subgroup of the orthogonal group O(d). Point groups are used to describe the symmetries
Point_group
Sporadic simple group
Nick; Hughes, Sam (2019), "The character table of a sharply 5-transitive subgroup of the alternating group of degree 12", International Journal of Group
Mathieu_group_M11
the derived group B2(2)′. A8 is isomorphic to A3(2). Remarks: An index 2 subgroup of the symmetric group of permutations of n points when n > 1. Notation:
List_of_finite_simple_groups
childhood vaccination, and have sought to expand their influence from niche subgroups into national political debates. Ideas that later coalesced into anti-vaccine
Anti-vaccine_activism
CG(k) is the semidirect product of a normal Hall π' subgroup I(K) with CH(k). The Feit–Thompson proof of the odd-order theorem uses "tamely embedded
Dade_isometry
248-dimensional exceptional simple Lie group
& Ryba (1999). The Dempwolff group is a subgroup of (the compact form of) E8. It is contained in the Thompson sporadic group, which acts on the underlying
E8_(mathematics)
Indigenous people of North America
south of Thompson. The O-Pipon-Na-Piwin Cree Nation is located in the settlement of South Indian Lake, 130 km (81 mi) northwest of Thompson. Marcel Colomb
Cree
Country in East Asia
indigenous people, the Taivoan, who were historically classified as a subgroup of the Siraya. Use of the current Chinese name (臺灣 / 台灣) became official
Taiwan
Country in South America
arrival of the Europeans, the boundaries between these groups and their subgroups were marked by wars that arose from differences in culture, language and
Brazil
Armed conflict in the Middle East
Affairs. 92 (3): 647–663. doi:10.1111/1468-2346.12599. Jones, Seth G.; Thompson, Jared; Ngo, Danielle; Bermudez, Joseph S. Jr.; McSorley, Brian (21 December
Yemeni civil war (2014–present)
Yemeni_civil_war_(2014–present)
Algebraic variety
quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z). The term modular
Modular_curve
From an exceptional automorphism of a Dynkin diagram
Kleidman determined their maximal subgroups. The Ree groups of type 2G2 are exceptionally hard to characterize. Thompson studied this problem, and was able
Ree_group
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 theorem in group theory
solvable (i.e., has a solvable subgroup of finite index) or it contains a nonabelian free group (i.e., it has a subgroup isomorphic to the free group on
Tits_alternative
Type of finite group
p.169) in order to extend some of Thompson's results about odd groups to groups with dihedral Sylow 2-subgroups. If a group has trivial p′ core Op′(G)
P-constrained_group
Type of group
studied in ways analogous to a finite group. Sylow subgroups, Carter subgroups, and abelian subgroups of locally finite groups have been studied. The concept
Locally_finite_group
Sporadic simple group
over the field with 4 elements. This arises from the fact that Fi22 is a subgroup of 2E6(22). All the ordinary and modular character tables of Fi22 have
Fischer_group_Fi22
Indigenous peoples of Western Canada and the United States
Bouchard 1998a, p. 189 Thompson & Thompson 1996, p. 147 Thompson & Thompson 1996, p. 45 Thompson & Thompson 1996, p. 104 Thompson & Thompson 1996, p. 171 Okanogan
Interior_Salish_peoples
Classification system for symmetry groups in geometry
structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively
Coxeter_notation
English singer (born 1948)
mother Annie [...] coming from rare 'Romanichal' stock, a subgroup of the Romani people Thompson, Dave (1 September 2014). Robert Plant: The Voice That Sailed
Robert_Plant
San Francisco, California, bound for Sanak Island in the Sanak Islands subgroup of the Fox Islands group of the Aleutian Islands with 28 fisherman and
List_of_shipwrecks_in_1904
Westernmost region of the African continent
West African medieval state in present-day southeastern Nigeria and a subgroup of the Igbo people. The Kingdom of Nri was unusual in the history of world
West_Africa
leans into LGBTQ rights". ny1.com. New York 1. Retrieved October 19, 2025. Thompson, Abbie (May 22, 2025). "Zohran Mamdani Announces Protection Plan For LGBTQ+
2025 New York City mayoral election
2025_New_York_City_mayoral_election
American groups originally from Switzerland
The Swiss Amish (Swiss German: Schwyzer Amisch) are a subgroup of the Amish that emigrated to the United States mostly in the middle of the 19th century
Swiss_Amish
Chemical compound found in some species of mushrooms
(relative to 1.3 overall), but based on only one study for that dosing subgroup. This meta-analysis included both RCTs and prospective open-label studies
Psilocybin
Conservative political initiative in the United States
powers could be militarized under Project 2025. In 2024, a Project 2025 subgroup, the Border Security Workgroup, crafted plans for a new nationwide militarized
Project_2025
Concept in mathematical group theory
normal subgroup of G. Each normal subgroup of G is the intersection of the kernels of some of the irreducible characters of G. The commutator subgroup of
Character_theory
Sporadic simple group
3, thus length √6. It is thus a subgroup of C o 0 {\displaystyle \mathrm {Co} _{0}} . It is isomorphic to a subgroup of C o 1 {\displaystyle \mathrm {Co}
Conway_group_Co3
and 1.8% of the total population. While Hui people are the most numerous subgroup, the greatest concentration of Muslims is in Xinjiang, which has a significant
Religion_in_China
theorem that all groups of odd order are solvable. Suppose that H is a subgroup of a finite group G, and S a set of irreducible characters of H. Write
Coherent_set_of_characters
Group of traditionalist Christian church fellowships
the Old Order Amish, though there are other subgroups of Amish. The Amish fall into three main subgroups—the Old Order Amish, the New Order Amish, and
Amish
Belgian mathematician (1930–2021)
a finitely generated subgroup of a linear group, then either G has a solvable subgroup of finite index or it has a free subgroup of rank 2. The Tits group
Jacques_Tits
Topics referred to by the same term
(Marvel Comics), a Marvel Comics title during the 1980s nWo Wolfpac, a subgroup within the New World Order professional wrestling stable Teenage Wolfpack
Wolfpack
travel, tourism, insurance
THOMPSON SUBGROUP
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travel, tourism, insurance