theory of groups is a part of mathematics which examines how groups act on given structures. Here the focus is in particular on operations of groups on...
105 KB (21,294 words) - 10:21, 1 April 2025
aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. As a consequence...
15 KB (1,831 words) - 16:54, 2 February 2025
mathematical finite group theory, an N-group is a group all of whose local subgroups (that is, the normalizers of nontrivial p-subgroups) are solvable groups. The...
7 KB (913 words) - 09:43, 24 March 2025
{\displaystyle P} . In this case the theory of automorphisms of a finite cyclic group can be used. Another special case is when n {\displaystyle n} is...
36 KB (5,264 words) - 19:51, 13 June 2025
mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points...
22 KB (2,985 words) - 04:28, 23 November 2024
In group theory, a discipline within abstract algebra, a special group is a finite group of prime power order that is either elementary abelian itself...
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complete classification of finite simple groups. Group theory has three main historical sources: number theory, the theory of algebraic equations, and...
39 KB (5,086 words) - 18:26, 11 April 2025
classification of finite simple groups, there are a number of groups which do not fit into any infinite family. These are called the sporadic simple groups, or the...
52 KB (2,079 words) - 11:12, 25 May 2025
classification of finite simple groups (popularly called the enormous theorem) is a result of group theory stating that every finite simple group is either cyclic...
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combinatorial group theory. A presentation is said to be finitely generated if S is finite and finitely related if R is finite. If both are finite it is said...
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representation theory (that is, through the representations of the group) and of computational group theory. A theory has been developed for finite groups, which...
103 KB (13,241 words) - 14:14, 11 June 2025
In the mathematical classification of finite simple groups, a thin group is a finite group such that for every odd prime number p, the Sylow p-subgroups...
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In the theory of algebraic groups, a special group is a linear algebraic group G with the property that every principal G-bundle is locally trivial in...
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automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The...
46 KB (6,212 words) - 09:18, 3 June 2025
group theory, a nilpotent group G is a group that has an upper central series that terminates with G. Equivalently, it has a central series of finite...
15 KB (1,912 words) - 08:01, 24 April 2025
known as group theory, the term Z-group refers to a number of distinct types of groups: in the study of finite groups, a Z-group is a finite group whose...
7 KB (889 words) - 01:36, 13 November 2023
study the group. It is mostly of interest for the study of infinite groups. Special cases of groups with finiteness properties are finitely generated...
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of finite simple groups says that most finite simple groups arise as the group G(k) of k-rational points of a simple algebraic group G over a finite field...
56 KB (8,018 words) - 09:30, 15 April 2025
representation theory; this special case has very different properties. See Representation theory of finite groups. Compact groups or locally compact groups — Many...
15 KB (2,245 words) - 02:55, 11 May 2025
of a finite group is the number of its elements. If a group is not finite, one says that its order is infinite. The order of an element of a group (also...
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quotient space is finite. crystallographic point group congruence subgroup arithmetic group geometric group theory computational group theory freely discontinuous...
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Minkowski's theorem Topological group Field Finite field Galois theory Grothendieck group Group ring Group with operators Heap Linear algebra Magma Module...
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In mathematics, specifically in group theory, an elementary abelian group is an abelian group in which all elements other than the identity have the same...
8 KB (991 words) - 12:15, 19 May 2025
alternating group is the group of even permutations of a finite set. The alternating group on a set of n elements is called the alternating group of degree...
17 KB (1,539 words) - 05:01, 21 October 2024
FC-group A group is an FC-group if every conjugacy class of its elements has finite cardinality. finite group A finite group is a group of finite order, that...
25 KB (2,955 words) - 11:01, 14 January 2025
one can prove that the center of any non-trivial finite p-group is non-trivial. If the quotient group G/Z(G) is cyclic, G is abelian (and hence G = Z(G)...
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finite reflection group. In fact it turns out that most finite reflection groups are Weyl groups. Abstractly, Weyl groups are finite Coxeter groups,...
21 KB (3,256 words) - 23:36, 23 November 2024
affine group of an affine space over a finite field of prime order. Groups similar to Galois groups are (today) called permutation groups. The theory of permutation...
32 KB (3,571 words) - 09:09, 15 May 2025
model theory, a stable group is a group that is stable in the sense of stability theory. An important class of examples is provided by groups of finite Morley...
7 KB (751 words) - 12:11, 20 November 2023
field theory and throughout algebraic geometry. A complete group scheme over a field need not be commutative, however; for example, any finite group scheme...
20 KB (2,860 words) - 23:16, 5 March 2025