The radical of an algebraic group is the

identity component In mathematics, specifically group theory, the identity component of a group ''G'' refers to several closely related notions of the largest connected subgroup of ''G'' containing the identity element. In point set topology, the identity compo ...

of its maximal

normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...

solvable

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

. For example, the radical of the

general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...

\operatorname_n(K)

(for a field ''K'') is the subgroup consisting of scalar matrices, i.e. matrices

(a_)

with

a_ = \dots = a_

and

a_=0

for

i \ne j

. An

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ...

is called

semisimple In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...

if its radical is trivial, i.e., consists of the identity element only. The group

\operatorname_n(K)

is semi-simple, for example. The subgroup of unipotent elements in the radical is called the

unipotent radical In mathematics, a unipotent element ''r'' of a ring ''R'' is one such that ''r'' − 1 is a nilpotent element; in other words, (''r'' − 1)''n'' is zero for some ''n''. In particular, a square matrix ''M'' is a unipote ...

, it serves to define

reductive group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...

References

"Radical of a group"
Encyclopaedia of Mathematics Algebraic groups {{algebra-stub

See also

References