Dicyclic groupIn group theory, a dicyclic group (notation Dicn or Q4n, ) is a particular kind of non-abelian group of order 4n (n > 1). It is an extension of the cyclic group of order 2 by a cyclic group of order 2n, giving the name di-cyclic. In the notation of exact sequences of groups, this extension can be expressed as: More generally, given any finite abelian group with an order-2 element, one can define a dicyclic group.
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.
Modular representation theoryModular representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups over a field K of positive characteristic p, necessarily a prime number. As well as having applications to group theory, modular representations arise naturally in other branches of mathematics, such as algebraic geometry, coding theory, combinatorics and number theory.
Schur multiplierIn mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group of a group G. It was introduced by in his work on projective representations. The Schur multiplier of a finite group G is a finite abelian group whose exponent divides the order of G. If a Sylow p-subgroup of G is cyclic for some p, then the order of is not divisible by p. In particular, if all Sylow p-subgroups of G are cyclic, then is trivial.
Point groups in three dimensionsIn geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E(3) of all isometries. Symmetry groups of geometric objects are isometry groups. Accordingly, analysis of isometry groups is analysis of possible symmetries.
Group ringIn algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring.
Binary tetrahedral groupIn mathematics, the binary tetrahedral group, denoted 2T or , is a certain nonabelian group of order 24. It is an extension of the tetrahedral group T or (2,3,3) of order 12 by a cyclic group of order 2, and is the of the tetrahedral group under the 2:1 covering homomorphism Spin(3) → SO(3) of the special orthogonal group by the spin group. It follows that the binary tetrahedral group is a discrete subgroup of Spin(3) of order 24. The complex reflection group named 3(24)3 by G.C.
BiquaternionIn abstract algebra, the biquaternions are the numbers w + x i + y j + z k, where w, x, y, and z are complex numbers, or variants thereof, and the elements of {1, i, j, k} multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof: Biquaternions when the coefficients are complex numbers. Split-biquaternions when the coefficients are split-complex numbers. Dual quaternions when the coefficients are dual numbers.
Split-biquaternionIn mathematics, a split-biquaternion is a hypercomplex number of the form where w, x, y, and z are split-complex numbers and i, j, and k multiply as in the quaternion group. Since each coefficient w, x, y, z spans two real dimensions, the split-biquaternion is an element of an eight-dimensional vector space. Considering that it carries a multiplication, this vector space is an algebra over the real field, or an algebra over a ring where the split-complex numbers form the ring.
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.
Characteristic subgroupIn mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphism, every characteristic subgroup is normal; though the converse is not guaranteed. Examples of characteristic subgroups include the commutator subgroup and the center of a group. A subgroup H of a group G is called a characteristic subgroup if for every automorphism φ of G, one has φ(H) ≤ H; then write H char G.
Quasidihedral groupIn mathematics, the quasi-dihedral groups, also called semi-dihedral groups, are certain non-abelian groups of order a power of 2. For every positive integer n greater than or equal to 4, there are exactly four isomorphism classes of non-abelian groups of order 2n which have a cyclic subgroup of index 2. Two are well known, the generalized quaternion group and the dihedral group. One of the remaining two groups is often considered particularly important, since it is an example of a 2-group of maximal nilpotency class.
Central seriesIn mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a central series means it is a nilpotent group; for matrix rings (considered as Lie algebras), it means that in some basis the ring consists entirely of upper triangular matrices with constant diagonal. This article uses the language of group theory; analogous terms are used for Lie algebras.
Special linear groupIn mathematics, the special linear group SL(n, F) of degree n over a field F is the set of n × n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant where F× is the multiplicative group of F (that is, F excluding 0). These elements are "special" in that they form an algebraic subvariety of the general linear group – they satisfy a polynomial equation (since the determinant is polynomial in the entries).
Nilpotent groupIn mathematics, specifically group theory, a nilpotent group G is a group that has an upper central series that terminates with G. Equivalently, its central series is of finite length or its lower central series terminates with {1}. Intuitively, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute. It is also true that finite nilpotent groups are supersolvable.
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.
Fundamental theorem of Galois theoryIn mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his development of Galois theory. In its most basic form, the theorem asserts that given a field extension E/F that is finite and Galois, there is a one-to-one correspondence between its intermediate fields and subgroups of its Galois group. (Intermediate fields are fields K satisfying F ⊆ K ⊆ E; they are also called subextensions of E/F.
VersorIn mathematics, a versor is a quaternion of norm one (a unit quaternion). Each versor has the form where the r2 = −1 condition means that r is a unit-length vector quaternion (or that the first component of r is zero, and the last three components of r are a unit vector in 3 dimensions). The corresponding 3-dimensional rotation has the angle 2a about the axis r in axis–angle representation. In case a = π/2 (a right angle), then , and the resulting unit vector is termed a right versor.
Cayley graphIn mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley), and uses a specified set of generators for the group. It is a central tool in combinatorial and geometric group theory. The structure and symmetry of Cayley graphs makes them particularly good candidates for constructing families of expander graphs.
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).
