In 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 quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure. For example, the cyclic group of addition modulo n can be obtained from the group of integers under addition by identifying elements that differ by a multiple of and defining a group structure that operates on each such class as a single entity. It is part of the mathematical field known as group theory.
In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic. From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished.
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.
In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups.
In mathematics, specifically abstract algebra, the isomorphism theorems are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product:
In 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.
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset of the quaternions under multiplication. It is given by the group presentation
In mathematics, group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group G in an associated G-moduleM to elucidate the properties of the group. By treating the G-module as a kind of topological space with elements of representing n-simplices, topological properties of the space may be computed, such as the set of cohomology groups . The cohomology groups in turn provide insight into the structure of the group G and G-module M themselves. Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper. As in algebraic topology, there is a dual theory called group homology. The techniques of group cohomology can also be extended to the case that instead of a G-module, G acts on a nonabelian G-group; in effect, a generalization of a module to non-Abelian coefficients.
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If and are two groups, then is an extension of by if there is a short exact sequence
In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “universal” group having these properties, in the sense that any two homomorphisms from G and H into a group K factor uniquely through a homomorphism from G ∗ H to K. Unless one of the groups G and H is trivial, the free product is always infinite. The construction of a free product is similar in spirit to the construction of a free group.
In mathematics, specifically in symplectic geometry, the momentum map is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The momentum map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including symplectic (Marsden–Weinstein) quotients, discussed below, and symplectic cuts and sums.
In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature. More precisely, they are two equivalent representations of the spin groups, which are double covers of the special orthogonal groups. They are usually studied over the real or complex numbers, but they can be defined over other fields.
In linear algebra, particularly projective geometry, a semilinear map between vector spaces V and W over a field K is a function that is a linear map "up to a twist", hence semi-linear, where "twist" means "field automorphism of K". Explicitly, it is a function T : V → W that is:
In mathematics, and more precisely in topology, the mapping class group of a surface, sometimes called the modular group or Teichmüller modular group, is the group of homeomorphisms of the surface viewed up to continuous deformation. It is of fundamental importance for the study of 3-manifolds via their embedded surfaces and is also studied in algebraic geometry in relation to moduli problems for curves.
In abstract algebra, a branch of mathematics, an affine monoid is a commutative monoid that is finitely generated, and is isomorphic to a submonoid of a free abelian group . Affine monoids are closely connected to convex polyhedra, and their associated algebras are of much use in the algebraic study of these geometric objects.
In mathematics, specifically group theory, finite groups of prime power order , for a fixed prime number and varying integer exponents , are briefly called finitep-groups.
In the mathematical subject geometric group theory, a fully irreducible automorphism of the free group Fn is an element of Out(Fn) which has no periodic conjugacy classes of proper free factors in Fn. Fully irreducible automorphisms are also referred to as "irreducible with irreducible powers" or "iwip" automorphisms. The notion of being fully irreducible provides a key Out(Fn) counterpart of the notion of a pseudo-Anosov element of the mapping class group of a finite type surface. Fully irreducibles play an important role in the study of structural properties of individual elements and of subgroups of Out(Fn).
In the mathematical subject of group theory, a co-Hopfian group is a group that is not isomorphic to any of its proper subgroups. The notion is dual to that of a Hopfian group, named after Heinz Hopf.
