Finite groupIn abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part of group theory since it arose in the 19th century.
Group theoryIn abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.
Simple groupIn mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgroup and the corresponding quotient group. This process can be repeated, and for finite groups one eventually arrives at uniquely determined simple groups, by the Jordan–Hölder theorem. The complete classification of finite simple groups, completed in 2004, is a major milestone in the history of mathematics.
Group (mathematics)In mathematics, a group is a non-empty set with an operation that satisfies the following constraints: the operation is associative, has an identity element, and every element of the set has an inverse element. Many mathematical structures are groups endowed with other properties. For example, the integers with the addition operation is an infinite group, which is generated by a single element called 1 (these properties characterize the integers in a unique way).
Group of Lie typeIn mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phrase group of Lie type does not have a widely accepted precise definition, but the important collection of finite simple groups of Lie type does have a precise definition, and they make up most of the groups in the classification of finite simple groups.
Symmetric groupIn abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group defined over a finite set of symbols consists of the permutations that can be performed on the symbols. Since there are ( factorial) such permutation operations, the order (number of elements) of the symmetric group is .
Monster groupIn the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order 2463205976112133171923293141475971 = 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 ≈ 8. The finite simple groups have been completely classified. Every such group belongs to one of 18 countably infinite families, or is one of 26 sporadic groups that do not follow such a systematic pattern.
Cyclic groupIn group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted Cn, that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as an integer power of g in multiplicative notation, or as an integer multiple of g in additive notation.
Outer automorphism groupIn mathematics, the outer automorphism group of a group, G, is the quotient, Aut(G) / Inn(G), where Aut(G) is the automorphism group of G and Inn(G) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out(G). If Out(G) is trivial and G has a trivial center, then G is said to be complete. An automorphism of a group that is not inner is called an outer automorphism. The cosets of Inn(G) with respect to outer automorphisms are then the elements of Out(G); this is an instance of the fact that quotients of groups are not, in general, (isomorphic to) subgroups.
Solvable groupIn mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup. Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equation. Specifically, a polynomial equation is solvable in radicals if and only if the corresponding Galois group is solvable (note this theorem holds only in characteristic 0).
Projective linear groupIn mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space V on the associated projective space P(V). Explicitly, the projective linear group is the quotient group PGL(V) = GL(V)/Z(V) where GL(V) is the general linear group of V and Z(V) is the subgroup of all nonzero scalar transformations of V; these are quotiented out because they act trivially on the projective space and they form the kernel of the action, and the notation "Z" reflects that the scalar transformations form the center of the general linear group.
Sylow theoremsIn mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups. For a prime number , a Sylow p-subgroup (sometimes p-Sylow subgroup) of a group is a maximal -subgroup of , i.
Character theoryIn mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form. Georg Frobenius initially developed representation theory of finite groups entirely based on the characters, and without any explicit matrix realization of representations themselves.
P-groupIn mathematics, specifically group theory, given a prime number p, a p-group is a group in which the order of every element is a power of p. That is, for each element g of a p-group G, there exists a nonnegative integer n such that the product of pn copies of g, and not fewer, is equal to the identity element. The orders of different elements may be different powers of p. Abelian p-groups are also called p-primary or simply primary. A finite group is a p-group if and only if its order (the number of its elements) is a power of p.
