field is reductive if it has a representation that has a finite kernel and is a direct sum of irreducible representations. Reductive groups include some...

reductive groups, but over non-perfect fields Jacques Tits found some examples of pseudo-reductive groups that are not reductive. A pseudo-reductive k-group...

require reductive groups to be connected.) A semisimple group is reductive. A group G over an arbitrary field k is called semisimple or reductive if G k...

a semidirect product of a unipotent group (its unipotent radical) with a reductive group. In turn reductive groups are decomposed as (again essentially)...

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...

between automorphic representations of a reductive group and homomorphisms from a Langlands group to an L-group. This offers numerous variations, in part...

the unipotent radical, it serves to define reductive groups. Reductive group Unipotent group "Radical of a group", Encyclopaedia of Mathematics v t e...

The Poincaré group, named after Henri Poincaré (1905), was first defined by Hermann Minkowski (1908) as the isometry group of Minkowski spacetime. It...

In mathematics, given two groups, (G,∗) and (H, ·), a group homomorphism from (G,∗) to (H, ·) is a function h : G → H such that for all u and v in G it...

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...

mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not...

In mathematics, the Klein four-group is an abelian group with four elements, in which each element is self-inverse (composing it with itself produces...

In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known...

a quasi-split group over a field is a reductive group with a Borel subgroup defined over the field. Simply connected quasi-split groups over a field correspond...

mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest...

A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that...

In mathematics, a permutation group is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations...

In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time...

In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to...

abelian normal subgroup of U(n), the unitary group is not semisimple, but it is reductive. The unitary group U(n) is endowed with the relative topology...

finite group is the number of its elements. If a group is not finite, one says that its order is infinite. The order of an element of a group (also called...

In mathematics and group theory, the term multiplicative group refers to one of the following concepts: the group under multiplication of the invertible...

In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension...

linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because...

In abstract algebra, a cyclic group or monogenous group is a group, denoted Cn (also frequently Z {\displaystyle \mathbb {Z} } n or Zn, not to be confused...

In mathematics, a Lie group (pronounced /liː/ LEE) is a group that is also a differentiable manifold, such that group multiplication and taking inverses...

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...

Reductive amination (also known as reductive alkylation) is a form of amination that converts a carbonyl group to an amine via an intermediate imine. The...

ISBN 978-3-540-07686-5, MR 0444786 Haboush, W. J. (1975), "Reductive groups are geometrically reductive", Annals of Mathematics, 102 (1): 67–83, doi:10.2307/1970974...

In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some...