Appendix:Glossary of group theory

This is a glossary of group theory. Throughout the article, we use e to denote the identity element of a group.

A

 * abelian group : A group $$(G,*)$$ is abelian if $$*$$ is commutative, i.e. $$g*h = h*g$$ for all $$g,h \in G$$. Likewise, a group is nonabelian if this relation fails to hold for any pair $$g,h \in G$$.

C

 * coset : If $$G$$ is a group, and $$H$$ is a subgroup of $$G$$, and $$g$$ is an element of $$G$$, then gH = {&thinsp;gh : h &isin; H&thinsp;} is the left coset of H in G with respect to g, and Hg = {&thinsp;hg : h &isin; H&thinsp;} is the right coset of H in G with respect to g.

D

 * 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.

F

 * 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&isin;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.


 * finitely generated group : If there exists a finite set S such that  = 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.


 * 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.

G

 * 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 : A set together with an associative operation which admits an identity element and such that every element has an inverse.


 * group isomorphism : Group homomorphisms that have inverse functions. The inverse of an isomorphism, it turns out, must also be a homomorphism.


 * group homomorphism : These are functions f : (G,*) &rarr; (H,&times;) that have the special property that f(a * b) = f(a) &times; f(b) for any elements a and b of G.


 * group representation : (not to be confused with the presentation of a group). 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.

I

 * 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.

K

 * kernel : Given a group homomorphism, the preimage of the identity in the codomain of the group homomorphism. Every normal subgroup is the kernel of a group homomorphism and vice versa.

N

 * normal subgroup : H is a normal subgroup of G if for all g in G and h in H, g * h * g&minus;1 also belongs to H.

O

 * order : Of a group $$(G, *)$$, the cardinality (i.e. number of elements) of $$G$$. (A group with finite order is called a finite group.)


 * order : Of an element of a group. Suppose $$x \in G$$ 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.

P

 * p-group : If p is prime, then a p-group is a group G where for each element g&isin;G, there exists a positive integer m such that the order of g is pm. A finite group is a p-group if and only if its group order (i.e. the number of its elements) is itself a power of p.


 * p-subgroup : A subgroup which is also p-group. (The study of p-subgroups is the central object of the Sylow theorems.)

S

 * 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 over the second half of the 20th century, the finite simple groups have all been.


 * subgroup : Given group (G,*), a subset H which remains a group when the operation * is restricted to H. Given a set S of G. We denote by  the smallest subgroup of G containing S.