Glossary of group theory
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A group is a set together with an associative operation that admits an identity element and such that there exists an inverse for every element.
Throughout this glossary, we use eScript error: No such module "Check for unknown parameters". to denote the identity element of a group.
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<dt id="Script error: No such module "delink"." >abelian group
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<dt id="Script error: No such module "delink"." >ascendant subgroup
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<dt id="Script error: No such module "delink"." >automorphism
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<dt id="Script error: No such module "delink"." >center of a group
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<dt id="Script error: No such module "delink"." >centerless group
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<dt id="Script error: No such module "delink"." >central subgroup
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<dt id="Script error: No such module "delink"." >centralizer
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<dt id="Script error: No such module "delink"." >characteristic subgroup
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<dt id="Script error: No such module "delink"." >characteristically simple group
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<dt id="Script error: No such module "delink"." >class function
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<dt id="Script error: No such module "delink"." >class number
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<dt id="Script error: No such module "delink"." >commutator
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<dt id="Script error: No such module "delink"." >commutator subgroup
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<dt id="Script error: No such module "delink"." >complete group
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<dt id="Script error: No such module "delink"." >composition series
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<dt id="Script error: No such module "delink"." >conjugacy-closed subgroup
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<dt id="Script error: No such module "delink"." >conjugacy class
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<dt id="Script error: No such module "delink"." >conjugate elements
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<dt id="Script error: No such module "delink"." >conjugate subgroups
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<dt id="Script error: No such module "delink"." >contranormal subgroup
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<dt id="Script error: No such module "delink"." >cyclic group
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<dt id="Script error: No such module "delink"." >derived subgroup
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<dt id="Script error: No such module "delink"." >direct product
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<dt id="Script error: No such module "delink"." >exponent of a group
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F
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<dt id="Script error: No such module "delink"." >factor group
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<dt id="Script error: No such module "delink"." >FC-group
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<dt id="Script error: No such module "delink"." >finite group
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<dt id="Script error: No such module "delink"." >finitely generated group
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G
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<dt id="Script error: No such module "delink"." >generating set
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<dt id="Script error: No such module "delink"." >group automorphism
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<dt id="Script error: No such module "delink"." >group homomorphism
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<dt id="Script error: No such module "delink"." >group isomorphism
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<dt id="Script error: No such module "delink"." >homomorphism
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<dt id="Script error: No such module "delink"." >index of a subgroup
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<dt id="Script error: No such module "delink"." >isomorphism
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<dt id="Script error: No such module "delink"." >lattice of subgroups
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<dt id="Script error: No such module "delink"." >locally cyclic group
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<dt id="Script error: No such module "delink"." >no small subgroup
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<dt id="Script error: No such module "delink"." >normal closure
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<dt id="Script error: No such module "delink"." >normal core
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<dt id="Script error: No such module "delink"." >normal series
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<dt id="Script error: No such module "delink"." >normal subgroup
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<dt id="Script error: No such module "delink"." >normalizer
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<dt id="Script error: No such module "delink"." >orbit
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<dt id="Script error: No such module "delink"." >order of a group
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<dt id="Script error: No such module "delink"." >order of a group element
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<dt id="Script error: No such module "delink"." >perfect core
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<dt id="Script error: No such module "delink"." >perfect group
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<dt id="Script error: No such module "delink"." >periodic group
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<dt id="Script error: No such module "delink"." >permutation group
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<dt id="Script error: No such module "delink"." >p-group
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<dt id="Script error: No such module "delink"." >p-subgroup
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<dt id="Script error: No such module "delink"." >quotient group
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<dt id="Script error: No such module "delink"." >real element
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<dt id="Script error: No such module "delink"." >serial subgroup
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<dt id="Script error: No such module "delink"." >simple group
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<dt id="Script error: No such module "delink"." >subgroup
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<dt id="Script error: No such module "delink"." >subgroup series
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<dt id="Script error: No such module "delink"." >subnormal subgroup
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<dt id="Script error: No such module "delink"." >symmetric group
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<dt id="Script error: No such module "delink"." >torsion group
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<dt id="Script error: No such module "delink"." >transitively normal subgroup
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<dt id="Script error: No such module "delink"." >trivial group
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Basic definitions
Both subgroups and normal subgroups of a given group form a complete lattice under inclusion of subsets; this property and some related results are described by the lattice theorem.
Kernel of a group homomorphism. It is the preimage of the identity in the codomain of a group homomorphism. Every normal subgroup is the kernel of a group homomorphism and vice versa.
Direct product, direct sum, and semidirect product of groups. These are ways of combining groups to construct new groups; please refer to the corresponding links for explanation.
Types of groups
Finitely generated group. If there exists a finite set SScript error: No such module "Check for unknown parameters". such that Template:Angbr = GScript error: No such module "Check for unknown parameters"., then GScript error: No such module "Check for unknown parameters". is said to be finitely generated. If SScript error: No such module "Check for unknown parameters". can be taken to have just one element, GScript error: No such module "Check for unknown parameters". is a cyclic group of finite order, an infinite cyclic group, or possibly a group Template:MsetScript error: No such module "Check for unknown parameters". with just one element.
Simple group. Simple groups are those groups having only eScript error: No such module "Check for unknown parameters". and themselves as normal subgroups. The name is misleading because a simple group can in fact be very complex. An example is the monster group, whose order is about 1054. Every finite group is built up from simple groups via group extensions, so the study of finite simple groups is central to the study of all finite groups. The finite simple groups are known and classified.
The structure of any finite abelian group is relatively simple; every finite abelian group is the direct sum of cyclic p-groups. This can be extended to a complete classification of all finitely generated abelian groups, that is all abelian groups that are generated by a finite set.
The situation is much more complicated for the non-abelian groups.
Free group. Given any set AScript error: No such module "Check for unknown parameters"., one can define a group as the smallest group containing the free semigroup of AScript error: No such module "Check for unknown parameters".. The group consists of the finite strings (words) that can be composed by elements from AScript error: No such module "Check for unknown parameters"., together with other elements that are necessary to form a group. Multiplication of strings is defined by concatenation, for instance (abb) • (bca) = abbbcaScript error: No such module "Check for unknown parameters"..
Every group (G, •)Script error: No such module "Check for unknown parameters". is basically a factor group of a free group generated by GScript error: No such module "Check for unknown parameters".. Refer to Presentation of a group for more explanation. One can then ask algorithmic questions about these presentations, such as:
- Do these two presentations specify isomorphic groups?; or
- Does this presentation specify the trivial group?
The general case of this is the word problem, and several of these questions are in fact unsolvable by any general algorithm.
General linear group, denoted by GL(n, F)Script error: No such module "Check for unknown parameters"., is the group of nScript error: No such module "Check for unknown parameters".-by-nScript error: No such module "Check for unknown parameters". invertible matrices, where the elements of the matrices are taken from a field FScript error: No such module "Check for unknown parameters". such as the real numbers or the complex numbers.
Group representation (not to be confused with the presentation of a group). A group representation is a homomorphism from a group to a general linear group. One basically tries to "represent" a given abstract group as a concrete group of invertible matrices, which is much easier to study.