Character of a group
A homomorphism of the given group into some standard Abelian group . Usually,
is taken to be either the multiplicative group
of a field
or the subgroup
![]() |
of . The concept of a character of a group was originally introduced for finite groups
with
(in this case every character
takes values in
).
The study of characters of groups reduces to the case of Abelian groups, since there is a natural isomorphism between the groups and
, where
is the commutator subgroup of
. The characters
form a linearly independent system in the space of all
-valued functions on
. A character
extends uniquely to a character of the group algebra
. The characters
are one-dimensional linear representations of
over
; the concept of a character of a representation of a group coincides in the one-dimensional case with the concept of a character of a group. Sometimes characters of a group are understood to mean characters of any of its finite-dimensional representations (and even to mean the representations themselves).
A character of a topological group is a continuous homomorphism
. If
is a locally compact Abelian group, then its characters separate points, that is, for any
,
, there exists a character
such that
. For Hausdorff Abelian groups
this assertion is not true, in general (see [3]). A character of an algebraic group
over an algebraically closed field
is a rational homomorphism
.
In number theory an important role is played by the characters of the multiplicative group of the residue ring
modulo
, which correspond one-to-one to Dirichlet characters modulo
: To a character
there corresponds the Dirichlet character
given by the formula
![]() |
See also Character group.
References
[1] | A. Borel, "Linear algebraic groups" , Benjamin (1969) |
[2] | S.A. Morris, "Pontryagin duality and the structure of locally compact Abelian groups" , London Math. Soc. Lecture Notes , 29 , Cambridge Univ. Press (1977) |
[3] | E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 1 , Springer (1963) |
Character of a group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Character_of_a_group&oldid=16966