Glossary of group theory

A group is a set together with an associative operation which admits an identity element and such that every element has an inverse.
Throughout the article, we use formula_1 to denote the identity element of a group.
D.
 
Basic definitions.
Subgroup. A subset formula_2 of a group formula_3 which remains a group when the operation formula_4 is restricted to formula_2 is called a "subgroup" of formula_6.
Given a subset formula_7 of formula_6. We denote by formula_9 the smallest subgroup of formula_6 containing formula_7. formula_9 is called the subgroup of formula_6 generated by formula_7.
Normal subgroup. formula_2 is a "normal subgroup" of formula_6 if for all formula_17 in formula_6 and formula_19 in formula_2, formula_21also belongs to formula_2.
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.
Group homomorphism. These are functions formula_23 that have the special property that
for any elements formula_25 and formula_26 of formula_6.
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.
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.
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 formula_7 such that formula_29 then formula_6 is said to be finitely generated. If formula_7 can be taken to have just one element, formula_6 is a cyclic group of finite order, an infinite cyclic group, or possibly a group formula_33 with just one element.
Simple group. Simple groups are those groups having only formula_1 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 formula_35, one can define a group as the smallest group containing the free semigroup of formula_35. The group consists of the finite strings (words) that can be composed by elements from formula_35, together with other elements that are necessary to form a group. Multiplication of strings is defined by concatenation, for instance formula_38
Every group formula_3 is basically a factor group of a free group generated by formula_6. Please refer to presentation of a group for more explanation.
One can then ask algorithmic questions about these presentations, such as:
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"), is the group of formula_41-by-formula_41 invertible matrices, where the elements of the matrices are taken from a field formula_43 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.