View source for Character group

{{MSC|20}}
{{TEX|done}}



The ''character group of a group $G$''
is the group $X(G) = \def\Hom{\textrm{Hom}}\Hom(G,A)$ of all characters of $G$ (cf.
[[Character of a group|Character of a group]]) with values in an Abelian group $A$, under the operation

$$\def\a{\alpha}\def\b{\beta}(\a\b)(g) = \a(g)\b(g),\quad g\in G,\quad \a,\b\in X(G),$$
induced by the operation in $A$. When $A=T = \{ z\in\C \;|\ \  |z| =1\}$, then

$$X(G) \simeq \prod_p \Hom(G,\Z(p^\infty)),$$
where $\Z(p^\infty)$ are [[Quasi-cyclic group | quasi-cyclic groups]], one for each prime number $p$. This group is algebraically compact (see
[[Pure submodule|Pure submodule]]). If $G$ is Abelian, then $X(G)$ is a
[[Divisible group|divisible group]] if and only if $G$ is torsion free and it is a reduced group if and only if $G$ is a [[Torsion group | torsion group]]
{{Cite|Fu}}.

The character group of a topological group $G$ is the group $X(G)$ of all continuous homomorphisms $G\to T$, equipped with the [[Compact-open_topology| compact-open topology]]. It is a Hausdorff Abelian topological group. If $G$ is locally compact, then so is $X(G)$; if $G$ is compact, then $X(G)$ is discrete, and if $G$ is discrete, then $X(G)$ is compact.

Examples of character groups:

$$X(T)\simeq \Z,\quad X(\Z)\simeq T,\quad X(\R) \simeq \R,\quad X(G)\simeq G $$
for any finite discrete Abelian group $G$.

With every continuous homomorphism of topological groups $\def\phi{\varphi}\phi:G\to H$ there is associated a homomorphism of the character groups $\phi^*:X(H)\to X(G)$. Here the correspondence $G\mapsto X(G)$, $\phi\mapsto\phi^*$, is a contravariant functor from the category of topological groups into the category of Abelian topological groups. If the category is restricted to locally compact Abelian groups $G$, then this functor determines an equivalence of that category and its dual category (see
[[Pontryagin duality|Pontryagin duality]]).

The character group of an [[Algebraic group | algebraic group]] $G$ over a field $K$ is the group $X(G)$ of all rational characters $\def\G{\mathbb{G}}G\to K^* = \G_m$. If $X(G)$ is an Abelian affine algebraic group, then $K[G]$ generates the space $G$ (that is, is a basis in this space) if and only if $G$ is a
[[Diagonalizable algebraic group|diagonalizable algebraic group]], i.e. is isomorphic to a closed subgroup of a certain [[Algebraic torus | algebraic torus]] $\G_m^s$. Here $X(G)$ is a finitely generated Abelian group (without $p$-torsion if $\def\char{\textrm{char}\;}\char K = p > 0$), and $K[G]$ is the [[group algebra]] of $X(G) $ over $K$, which makes it possible to define a duality between the categories of diagonalizable groups and that of finitely generated Abelian groups (without $p$-torsion if $\char K = p > 0$), cf.
{{Cite|Bo}}. When $G$ is a finite group (regarded as a $0$-dimensional algebraic group) and $\char K = 0$, then this duality is the same as the classical
[[Duality|duality]] of finite Abelian groups.

For any connected algebraic group $G$, the group $X(G) $ is torsion free. In particular, a diagonalizable group $G$ is a torus if and only if $X(G)\simeq \Z^s$.



====Comments====
An Abelian group is reduced if it contains no non-trivial divisible subgroups.

Above, the phrase  "character"  is of course strictly used in its narrowest meaning of a (continuous) homomorphism $G\to T$ and not in the sense of the character of some representation.

The character groups of many locally Abelian groups can be found in
{{Cite|HeRo}}.


====References====
{|
|-
|valign="top"|{{Ref|Bo}}||valign="top"|  A. Borel,   "Linear algebraic groups", Benjamin  (1969)  {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}}
|-
|valign="top"|{{Ref|Bo2}}||valign="top"|  N. Bourbaki,   "Elements of mathematics. Spectral theories", Addison-Wesley  (1977)  (Translated from French)  {{MR|0583191}} {{ZBL|1106.46004}}
|-
|valign="top"|{{Ref|Fu}}||valign="top"|  L. Fuchs,   "Infinite abelian groups", '''1''', Acad. Press  (1970)  {{MR|0255673}} {{ZBL|0209.05503}}
|-
|valign="top"|{{Ref|HeRo}}||valign="top"|  E. Hewitt,   K.A. Ross,   "Abstract harmonic analysis", '''1''', Springer  (1963)  {{MR|0156915}} {{ZBL|0115.10603}}
|-
|valign="top"|{{Ref|Hu}}||valign="top"|  J.E. Humphreys,   "Linear algebraic groups", Springer  (1975)  {{MR|0396773}} {{ZBL|0325.20039}}
|-
|valign="top"|{{Ref|Mo}}||valign="top"|  S.A. Morris,   "Pontryagin duality and the structure of locally compact Abelian groups", ''London Math. Soc. Lecture Notes'', '''29''', Cambridge Univ. Press  (1977)  {{MR|0442141}} {{ZBL|0446.22006}}
|-
|valign="top"|{{Ref|Po}}||valign="top"|  L.S. Pontryagin,   "Topological groups", Princeton Univ. Press  (1958)  (Translated from Russian)  {{MR|0201557}} {{ZBL|0022.17104}}
|-
|}