Glossary of group theory

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In group theory, a group (G, •) is a set G closed under a binary operation • satisfying the following 3 axioms:

  • Associativity: For all a, b and c in G, (ab) • c = a • (bc).
  • Identity element: There exists an eG such that for all a in G, ea = ae = a.
  • Inverse element: For each a in G, there is an element b in G such that ab = ba = e, where e is an identity element.

Basic examples for groups are the integers Z with addition operation, or rational numbers without zero Q \ {0} with multiplication. More generally, for any ring R, the units in R form a multiplicative group. See the group article for an illustration of this definition and for further examples. Groups include, however, much more general structures than the above. Group theory is concerned with proving abstract statements about groups, regardless of the actual nature of element and the operation of the groups in question.

This glossary provides short explanations of some basic notions used throughout group theory. Please refer to group theory for a general description of the topic. See also list of group theory topics.

Basic definitions[edit]

A subset HG is a subgroup if the restriction of • to H is a group operation on H. It is called normal, if left and right cosets agree, i.e. gH = Hg for all g in G. Normal subgroups play a distinguished role by virtue of the fact that the collection of cosets of a normal subgroup N in a group G naturally inherits a group structure, enabling the formation of the quotient group, usually denoted G/N (also called a factor group). The butterfly lemma is a technical result on the lattice of subgroups of a group.

Given a subset S of a group G, the smallest subgroup of G containing S is called the subgroup generated by S. It is often denoted <S>.

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.

Given any set A, one can define a group as the smallest group containing the free semigroup of A. This group consists of the finite strings called words that can be composed by elements from A and their inverses. 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 by the set of its elements. This phenomenon is made formal with group presentations.

The direct product, direct sum, and semidirect product of groups glue several groups together, in different ways. The direct product of a family of groups Gi, for example, is the cartesian product of the sets underlying the various Gi, and the group operation is performed component-wise.

A group homomorphism is a map f : GH between two groups that preserves the structure imposed by the operation, i.e.

f(ab) = f(a) • f(b).

Bijective (in-, surjective) maps are isomorphisms of groups (mono-, epimorphisms, respectively). The kernel ker(f) is always a normal subgroup of the group. For f as above, the fundamental theorem on homomorphisms relates the structure of G and H, and of the kernel and image of the homomorphism, namely

G / ker(f) ≅ im(f).

One of the fundamental problems of group theory is the classification of groups up to isomorphism.

Groups together with group homomorphisms form a category.

In universal algebra, groups are generally treated as algebraic structures of the form (G, •, e, −1), i.e. the identity element e and the map that takes every element a of the group to its inverse a−1 are treated as integral parts of the formal definition of a group.

Finiteness conditions[edit]

The order |G| (or o(G)) of a group is the cardinality of G. If the order |G| is (in-)finite, then G itself is called (in-)finite. An important class is the group of permutations or symmetric groups of N letters, denoted SN. Cayley's theorem exhibits any finite group G as a subgroup of the symmetric group on G. The theory of finite groups is very rich. Lagrange's theorem states that the order of any subgroup H of a finite group G divides the order of G. A partial converse is given by the Sylow theorems: if pn is the greatest power of a prime p dividing the order of a finite group G, then there exists a subgroup of order pn, and the number of these subgroups is also known. A projective limit of finite groups is called profinite.[1] An important profinite group, fundamental for p-adic analysis, class field theory, and l-adic cohomology is the ring of p-adic integers and the profinite completion of Z, respectively

\mathbb Z_p := \varprojlim_n \mathbb Z / p^n and \hat{\mathbb Z} := \varprojlim_n \mathbb Z / n.[2]

Most of the facts from finite groups can be generalized directly to the profinite case.[3]

Certain conditions on chains of subgroups, parallel to the notion of Noetherian and Artinian rings, allow to deduce further properties. For example the Krull-Schmidt theorem states that a group satisfying certain finiteness conditions for chains of its subgroups, can be uniquely written as a finite direct product of indecomposable subgroups.

Another, yet slightly weaker, level of finiteness is the following: a subset A of G is said to generate the group if any element h can be written as the product of elements of A. A group is said to be finitely generated if it is possible to find a finite subset A generating the group. Finitely generated groups are in many respects as well-treatable as finite groups.

Abelian groups[edit]

The category of groups can be subdivided in several ways. A particularly well-understood class of groups are the so-called abelian (in honor of Niels Abel) or commutative groups, i.e. the ones satisfying

\forall a,b \in G,\ a*b = b*a \mbox{.}

Another way of saying this is that the commutator

[a,b] =\,\! :\, a^{-1} b^{-1} ab

equals the identity element for all a and b. A non-abelian group is a group that is not abelian. Even more particular, cyclic groups are the groups generated by a single element. Being either isomorphic to Z or to Zn, the integers modulo n, they are always abelian. Any finitely generated abelian group is known to be a direct sum of groups of these two types. The category of abelian groups is an abelian category. In fact, abelian groups serve as the prototype of abelian categories. A converse is given by Mitchell's embedding theorem.

Normal series[edit]

Most of the notions developed in group theory are designed to tackle non-abelian groups. There are several notions designed to measure how far a group is from being abelian. The commutator subgroup (or derived group) is the subgroup generated by commutators [a, b], whereas the center is the subgroup of elements that commute with every other group element.

Given a group G and a normal subgroup N of G, denoted NG, there is an exact sequence:

1 → NGH → 1,

where 1 denotes the trivial group and H is the quotient G/N. This permits the decomposition of G into two smaller pieces. The other way round, given two groups N and H, a group G fitting into an exact sequence as above is called an extension of H by N. Given H and N there are many different group extensions G, which leads to the extension problem. There is always at least one extension, called the trivial extension, namely the direct sum 'G = N H, but usually there are more. For example, the Klein four-group is a non-trivial extension of Z2 by Z2. This is a first glimpse of homological algebra and Ext functors.[4]

Many properties for groups, for example being a finite group or a p-group (i.e. the order of every element is a power of p) are stable under extensions and sub- and quotient groups, i.e. if N and H have the property, then so does G and vice versa. This kind of information is therefore preserved while breaking it into pieces by means of exact sequences. If this process has come to an end, i.e. if a group G does not have any (non-trivial) normal subgroups, G is called simple. 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. The finite simple groups are known and classified.

Repeatedly taking normal subgroups (if they exist) leads to normal series:

1 = G0G1 ⊲ ... ⊲ Gn = G,

i.e. any Gi is a normal subgroup of the next one Gi+1. A group is solvable (or soluble) if it has a normal series all of whose quotients are abelian. Imposing further commutativity constraints on the quotients Gi+1 / Gi, one obtains central series which lead to nilpotent groups. They are an approximation of abelian groups in the sense that

[...[[g1g2], g3] ..., gn]=1

for all choices of group elements gi.

There may be distinct normal series for a group G. If it is impossible to refine a given series by inserting further normal subgroups, it is called composition series. By the Jordan–Hölder theorem any two composition series of a given group are equivalent.[5]

Other notions[edit]

General linear group, denoted by GL(n, F), is the group of n-by-n invertible matrices, where the elements of the matrices are taken from a field F 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.

Notes[edit]

  1. ^ Shatz 1972
  2. ^ These two groups play a central role for maximal abelian extension of number fields, see Kronecker–Weber theorem
  3. ^ For example the Sylow theorems.
  4. ^ Weibel 1994
  5. ^ This can be shown using the Schreier refinement theorem.

References[edit]