The center of a groupG, denoted Z(G), is the set of those group elements that commute with all elements of G, that is, the set of all h ∈ G such that hg = gh for all g ∈ G. Z(G) is always a normal subgroup of G. A group G is abelian if and only if Z(G) = G.
The commutator of two elements g and h 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 elements g and h is equal to the group's identity if and only if g and h commutate, that is, if and only if gh = hg.
with strict inclusions, such that each Hi is a maximal strict normal subgroup of Hi+1. Equivalently, a composition series is a subnormal series such that each factor groupHi+1 / Hi is simple. The factor groups are called composition factors.
Two elements x and y of a group G are conjugate if there exists an element g ∈ G such that g−1xg = y. The element g−1xg, denoted xg, is called the conjugate of x by g. Some authors define the conjugate of x by g as gxg−1. This is often denoted gx. Conjugacy is an equivalence relation. Its equivalence classes are called conjugacy classes.
Two subgroups H1 and H2 of a group G are conjugate subgroups if there is a g ∈ G such that gH1g−1 = H2.
A cyclic group is a group that is generated by a single element, that is, a group such that there is an element g 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.
The direct product of two groups G and H, denoted G × H, is the cartesian product of the underlying sets of G and H, 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.
A finite group is a group of finite order, that is, a group with a finite number of elements.
finitely generated group
A group G is finitely generated if there is a finite generating set, that is, if there is a finite set S of elements of G such that every element of G can be written as the combination of finitely many elements of S and of inverses of elements of S.
A generating set of a group G is a subset S of G such that every element of G can be expressed as a combination (under the group operation) of finitely many elements of S and inverses of elements of S.
Given two groups (G, ∗) and (H, ·), a homomorphism from G to H is a functionh : G → H such that for all a and b in G, h(a∗b) = h(a) · h(b).
index of a subgroup
The index of a subgroupH of a group G, denoted |G : H| or [G : H] or (G : H), is the number of cosets of H in G. For a normal subgroupN of a group G, the index of N in G is equal to the order of the quotient groupG / N. For a finite subgroup H of a finite group G, the index of H in G is equal to the quotient of the orders of G and H.
Given two groups (G, ∗) and (H, ·), an isomorphism between G and H is a bijectivehomomorphism from G to H, that is, a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. Two groups are isomorphic 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.
For a subset S of a group G, the normalizer of S in G, denoted NG(S), is the subgroup of G defined by
A normal series of a group G is a sequence of normal subgroups of G such that each element of the sequence is a normal subgroup of the next element:
A subgroupN of a group G is normal in G (denoted ) if the conjugation of an element n of N by an element g of G is always in N, that is, if for all g ∈ G and n ∈ N, gng−1 ∈ N. A normal subgroup N of a group G can be used to construct the quotient groupG/N (G mod N).
The order of an elementg of a group G is the smallest positiveintegern such that gn = e. If no such integer exists, then the order of g is said to be infinite. The order of a finite group is divisible by the order of every element.
An element g of a group G is called a real element of G if it belongs to the same conjugacy class as its inverse, that is, if there is a h in Gwith , where is defined as h−1gh. An element of a group G is real if and only if for all representations of G the trace of the corresponding matrix is a real number.
A subgroup of a group G is a subsetH of the elements of G that itself forms a group when equipped with the restriction of the group operation of G to H×H. A subset H of a group G is a subgroup of G if and only if it is nonempty and closed under products and inverses, that is, if and only if for every a and b in H, ab and a−1 are also in H.
A subgroup series of a group G is a sequence of subgroups of G such that each element in the series is a subgroup of the next element:
A subgroupH of a group G is a subnormal subgroup of G if there is a finite chain of subgroups of the group, each one normal in the next, beginning at H and ending at G.
Isomorphic groups. Two groups are isomorphic 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.
One of the fundamental problems of group theory is the classification of groupsup to isomorphism.
Simple group. Simple groups are those groups having only 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 , one can define a group as the smallest group containing the free semigroup of . The group consists of the finite strings (words) that can be composed by elements from , together with other elements that are necessary to form a group. Multiplication of strings is defined by concatenation, for instance
Every group is basically a factor group of a free group generated by . Please 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.
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.