A group(G, •) isabelian if• is commutative, i.e.g •h =h •g for allg,h ∈G. Likewise, a group isnonabelian if this relation fails to hold for any pairg,h ∈G.
ascendant subgroup
AsubgroupH of a groupG isascendant if there is an ascendingsubgroup series starting fromH and ending atG, such that every term in the series is anormal subgroup of its successor. The series may be infinite. If the series is finite, then the subgroup issubnormal.
Thecenter of a groupG, denotedZ(G), is the set of those group elements that commute with all elements ofG, that is, the set of allh ∈G such thathg =gh for allg ∈G.Z(G) is always anormal subgroup ofG. A group G isabelian if and only ifZ(G) =G.
Thecommutator of two elementsg andh of a group G is the element[g,h] =g−1h−1gh. Some authors define the commutator as[g,h] =ghg−1h−1 instead. The commutator of two elementsg andh is equal to the group's identity if and only ifg andh commutate, that is, if and only ifgh =hg.
with strict inclusions, such that eachHi is a maximal strictnormal subgroup ofHi+1. Equivalently, a composition series is a subnormal series such that eachfactor groupHi+1 /Hi issimple. The factor groups are called composition factors.
conjugacy-closed subgroup
Asubgroup of a group is said to beconjugacy-closed if any two elements of the subgroup that areconjugate in the group are also conjugate in the subgroup.
conjugacy class
Theconjugacy classes of a groupG are those subsets ofG containing group elements that areconjugate with each other.
conjugate elements
Two elementsx andy of a group G areconjugate if there exists an elementg ∈G such thatg−1xg =y. The elementg−1xg, denotedxg, is called the conjugate ofx byg. Some authors define the conjugate ofx byg asgxg−1. This is often denotedgx. Conjugacy is anequivalence relation. Itsequivalence classes are calledconjugacy classes.
conjugate subgroups
Two subgroupsH1 andH2 of a groupG areconjugate subgroups if there is ag ∈G such thatgH1g−1 =H2.
Acyclic group is a group that isgenerated by a single element, that is, a group such that there is an elementg in the group such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse.
Thedirect product of two groupsG andH, denotedG ×H, is thecartesian product of the underlying sets ofG andH, equipped with a component-wise defined binary operation(g1,h1) · (g2,h2) = (g1 ⋅g2,h1 ⋅h2). With this operation,G ×H itself forms a group.
The exponent of a groupG is the smallest positive integern such thatgn =e for allg ∈G. It is theleast common multiple of theorders of all elements in the group. If no such positive integer exists, the exponent of the group is said to be infinite.
Afinite group is a group of finiteorder, that is, a group with a finite number of elements.
finitely generated group
A groupG isfinitely generated if there is a finitegenerating set, that is, if there is a finite setS of elements ofG such that every element ofG can be written as the combination of finitely many elements ofS and of inverses of elements ofS.
Agenerating set of a groupG is a subsetS ofG such that every element ofG can be expressed as a combination (under the group operation) of finitely many elements ofS and inverses of elements ofS. Given a subsetS ofG. We denote by⟨S⟩ the smallest subgroup ofG containingS.⟨S⟩ is called the subgroup ofG generated byS.
Theindex of asubgroupH of a groupG, denoted|G :H| or[G :H] or(G :H), is the number ofcosets ofH inG. For anormal subgroupN of a groupG, the index ofN inG is equal to theorder of thequotient groupG /N. For afinite subgroupH of a finite groupG, the index ofH inG is equal to the quotient of the orders ofG andH.
isomorphism
Given two groups(G, •) and(H, ·), anisomorphism betweenG andH is abijectivehomomorphism fromG toH, that is, a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. Two groups areisomorphic if there exists a group isomorphism mapping from one to the other. Isomorphic groups can be thought of as essentially the same, only with different labels on the individual elements.
Anormal series of a group G is a sequence ofnormal subgroups ofG such that each element of the sequence is a normal subgroup of the next element:
with
.
normal subgroup
AsubgroupN of a groupG isnormal inG (denotedN ◅G) if theconjugation of an elementn ofN by an elementg ofG is always inN, that is, if for allg ∈G andn ∈N,gng−1 ∈N. A normal subgroupN of a groupG can be used to construct thequotient groupG /N.
normalizer
For a subsetS of a group G, thenormalizer ofS inG, denotedNG(S), is the subgroup ofG defined by
Consider a groupG acting on a setX. Theorbit of an elementx inX is the set of elements inX to whichx can be moved by the elements ofG. The orbit ofx is denoted byG ⋅x
Theorder of an elementg of a groupG is the smallestpositiveintegern such thatgn =e. If no such integer exists, then the order ofg is said to be infinite. The order of a finite group isdivisible by the order of every element.
Ifp is aprime number, then ap-group is one in which the order of every element is a power ofp. A finite group is ap-group if and only if theorder of the group is a power ofp.
An elementg of a groupG is called areal element ofG if it belongs to the sameconjugacy class as its inverse, that is, if there is ah inG with gh =g−1, wheregh is defined ash−1gh. An element of a groupG is real if and only if for allrepresentations ofG thetrace of the corresponding matrix is a real number.
AsubgroupH of a groupG is aserial subgroup ofG if there is a chainC of subgroups ofG fromH toG such that for each pair of consecutive subgroupsX andY inC,X is anormal subgroup ofY. If the chain is finite, thenH is asubnormal subgroup ofG.
Asubgroup of a groupG is asubsetH of the elements ofG that itself forms a group when equipped with the restriction of thegroup operation ofG toH ×H. A subsetH of a groupG is a subgroup ofG if and only if it is nonempty andclosed under products and inverses, that is, if and only if for everya andb inH,ab anda−1 are also inH.
subgroup series
Asubgroup series of a groupG is a sequence ofsubgroups ofG such that each element in the series is a subgroup of the next element:
subnormal subgroup
AsubgroupH of a groupG is asubnormal subgroup ofG if there is a finite chain of subgroups of the group, each onenormal in the next, beginning atH and ending atG.
Atrivial group is a group consisting of a single element, namely the identity element of the group. All such groups areisomorphic, and one often speaks ofthe trivial group.
Both subgroups and normal subgroups of a given group form acomplete lattice under inclusion of subsets; this property and some related results are described by thelattice theorem.
Kernel of a group homomorphism. It is thepreimage of the identity in thecodomain of a group homomorphism. Every normal subgroup is the kernel of a group homomorphism and vice versa.
Direct product,direct sum, andsemidirect product of groups. These are ways of combining groups to construct new groups; please refer to the corresponding links for explanation.
Simple group. Simple groups are those groups having onlye and themselves asnormal subgroups. The name is misleading because a simple group can in fact be very complex. An example is themonster group, whoseorder is about 1054. Every finite group is built up from simple groups viagroup extensions, so the study of finite simple groups is central to the study of all finite groups. The finite simple groups are known andclassified.
The structure of any finite abelian group is relatively simple; every finite abelian group is the direct sum ofcyclic p-groups.This can be extended to a complete classification of allfinitely generated abelian groups, that is all abelian groups that aregenerated by a finite set.
The situation is much more complicated for the non-abelian groups.
Free group. Given any setA, one can define a group as the smallest group containing thefree semigroup ofA. The group consists of the finite strings (words) that can be composed by elements fromA, together with other elements that are necessary to form a group. Multiplication of strings is defined by concatenation, for instance(abb) • (bca) =abbbca.
Every group(G, •) is basically a factor group of a free group generated byG. Refer toPresentation of a group for more explanation.One can then askalgorithmic 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 theword problem, and several of these questions are in fact unsolvable by any general algorithm.
General linear group, denoted byGL(n,F), is the group ofn-by-ninvertible matrices, where the elements of the matrices are taken from afieldF such as the real numbers or the complex numbers.
Group representation (not to be confused with thepresentation of a group). Agroup 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 invertiblematrices, which is much easier to study.
