Please refer to group theory for a general description of the topic. See also list of group theory topics.
A group is a set together with an associative operation such that every element has an inverse.
Throughout the article, we use e to denote the identity element of a group.
Table of contents |
2 Types of groups 3 Miscellaneous |
Basic definitions
Order of a group. Order of a group (G,*) is the cardinality (i.e. the size) of G. A group with finite order is called a finite group.
Order of an element of a group. Suppose x∈G and there exists a positive integer m such that x^{m} = e, then the smallest possible m is called the order of x. The order of a finite group is divisible by the order of every element.
Subgroup. A subset H of a group (G,*) which remains a group when the operation * is restricted to H is called a subgroup of G.
Given a set S of G. We denote by <S> the smallest subgroup of G containing S.
Normal subgroup. H is a normal subgroup G if for all g in G and h in H, g * h * g^{−1} also belongs to H.
Both subgroups and normal subgroups of given group form a complete lattice under inclusion of subsets; this property and some related results are described by the lattice theorem.
Group homomorphism. These are functions f : (G,*) → (H,×) that have the special property that
- f(a * b) = f(a) × f(b)
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 isomorphism and vice versa.
Group isomorphism. Group homomorphisms that have inverse functions. The inverse of an isomorphism, it turns out, must also be a homomorphism.
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 groups up to isomorphism.
Factor group, or quotient group. Given a group G and a normal subgroup N of G, the quotient group is the set G/N of left cosets {aN : a∈G} together with the operation aN*bN=abN. The relationship between normal subgroups, homomorphisms, and factor groups is summed up in the fundamental theorem on homomorphisms.
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
Abelian group. A group (G,*) is abelian if * is commutative, i.e. g*h=h*g for all g,h∈G.
Finitely generated group. If there exists a finite set S such that <S> = G, then G is said to be finitely generated. If S can be taken to have just one element, G is a cyclic group of finite order, an infinite cyclic group, or possibly a group {e} with just one element.
p-group. If p is prime, then a p-group is just a group with order p^{m} for some m.
p-subgroup. A subgroup which is also p-group.
The study of p-subgroups is the central object of the Sylow theorems.
Simple group. Simple groups are those groups with {e} and itself as the only normal subgroups. The name is misleading as its structure could be extremely complex. An example is the monster group, a group of order more than one million. Every finite group is built up from simple groups through the use of group extensions, so the study and classification of finite simple groups is central to the study of finite groups in general. As a result of extensive effort ove the second half of the 20th century, the finite simple groups have all been 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 A, one can define a multiplication of words as follows: (abb)*(bca)=abbbca. The free group generated by A is the smallest group containing this semigroup.
Every group (G,*) is basically a factor group of a free group generated by G. 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?
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.
Miscellaneous
Composition seriesNormal series