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A homomorphism of the given group into some standard Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c0215601.png" />. Usually, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c0215602.png" /> is taken to be either the multiplicative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c0215603.png" /> of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c0215604.png" /> or the subgroup
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{{TeX|done}}
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<math>\DeclareMathOperator\Hom{Hom}</math>
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A homomorphism of the given group into some standard Abelian group $A$. Usually, $A$ is taken to be either the multiplicative group $k^*$ of a field $k$ or the subgroup
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c0215605.png" /></td> </tr></table>
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\[ T = \{ z ∈ ℂ: |z| = 1 \} \]
  
of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c0215606.png" />. The concept of a character of a group was originally introduced for finite groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c0215607.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c0215608.png" /> (in this case every character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c0215609.png" /> takes values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156010.png" />).
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of $ℂ^*$. The concept of a character of a group was originally introduced for finite groups $G$ with $A = T$ (in this case every character $G → ℂ^*$ takes values in $T$).
  
The study of characters of groups reduces to the case of Abelian groups, since there is a natural isomorphism between the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156013.png" /> is the commutator subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156014.png" />. The characters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156015.png" /> form a linearly independent system in the space of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156016.png" />-valued functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156017.png" />. A character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156018.png" /> extends uniquely to a character of the group algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156019.png" />. The characters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156020.png" /> are one-dimensional linear representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156021.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156022.png" />; the concept of a [[Character of a representation of a group|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).
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The study of characters of groups reduces to the case of Abelian groups, since there is a natural isomorphism between the groups $\Hom(G, A)$ and $\Hom(G / (G, G), A)$, where $(G, G)$ is the commutator subgroup of $G$. The characters $G → k^*$ form a linearly independent system in the space of all $k$-valued functions on $G$. A character $G → k^*$ extends uniquely to a character of the group algebra $k[G]$. The characters $G → k^*$ are one-dimensional linear representations of $G$ over $k$; the concept of a [[Character of a representation of a group|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156023.png" /> is a continuous homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156024.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156025.png" /> is a locally compact Abelian group, then its characters separate points, that is, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156027.png" />, there exists a character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156028.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156029.png" />. For Hausdorff Abelian groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156030.png" /> this assertion is not true, in general (see [[#References|[3]]]). A character of an algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156031.png" /> over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156032.png" /> is a rational homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156033.png" />.
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A character of a topological group $G$ is a continuous homomorphism $G → T$. If $G$ is a locally compact Abelian group, then its characters separate points, that is, for any $a,b ∈ G$, $a ≠ b$, there exists a character : G → T$ such that $α(a) ≠ α(b)$. For Hausdorff Abelian groups $G$ this assertion is not true, in general (see [[#References|[3]]]). A character of an algebraic group $F$ over an algebraically closed field $K$ is a rational homomorphism $G → K^*$.
  
In number theory an important role is played by the characters of the multiplicative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156034.png" /> of the residue ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156035.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156036.png" />, which correspond one-to-one to Dirichlet characters modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156037.png" />: To a character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156038.png" /> there corresponds the [[Dirichlet character|Dirichlet character]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156039.png" /> given by the formula
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In number theory an important role is played by the characters of the multiplicative group $ℤ_k^*$ of the residue ring $ℤ_k$ modulo $k$, which correspond one-to-one to Dirichlet characters modulo $k$: To a character : ℤ_k^* → T$ there corresponds the [[Dirichlet character|Dirichlet character]] : ℤ → ℂ$ given by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021560/c02156040.png" /></td> </tr></table>
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\[ χ(n) = \begin{cases}
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α(n + kℤ) & \text{if}\quad (n, k) = 1;\\
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0        & \text{if}\quad (n, k) ≠ 1.
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\end{cases} \]
  
 
See also [[Character group|Character group]].
 
See also [[Character group|Character group]].

Revision as of 13:36, 23 July 2015

Template:TeX \(\DeclareMathOperator\Hom{Hom}\) A homomorphism of the given group into some standard Abelian group $A$. Usually, $A$ is taken to be either the multiplicative group $k^*$ of a field $k$ or the subgroup

\[ T = \{ z ∈ ℂ: |z| = 1 \} \]

of $ℂ^*$. The concept of a character of a group was originally introduced for finite groups $G$ with $A = T$ (in this case every character $G → ℂ^*$ takes values in $T$).

The study of characters of groups reduces to the case of Abelian groups, since there is a natural isomorphism between the groups $\Hom(G, A)$ and $\Hom(G / (G, G), A)$, where $(G, G)$ is the commutator subgroup of $G$. The characters $G → k^*$ form a linearly independent system in the space of all $k$-valued functions on $G$. A character $G → k^*$ extends uniquely to a character of the group algebra $k[G]$. The characters $G → k^*$ are one-dimensional linear representations of $G$ over $k$; 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 $G$ is a continuous homomorphism $G → T$. If $G$ is a locally compact Abelian group, then its characters separate points, that is, for any $a,b ∈ G$, $a ≠ b$, there exists a character $α: G → T$ such that $α(a) ≠ α(b)$. For Hausdorff Abelian groups $G$ this assertion is not true, in general (see [3]). A character of an algebraic group $F$ over an algebraically closed field $K$ is a rational homomorphism $G → K^*$.

In number theory an important role is played by the characters of the multiplicative group $ℤ_k^*$ of the residue ring $ℤ_k$ modulo $k$, which correspond one-to-one to Dirichlet characters modulo $k$: To a character $α: ℤ_k^* → T$ there corresponds the Dirichlet character $χ: ℤ → ℂ$ given by the formula

\[ χ(n) = \begin{cases} α(n + kℤ) & \text{if}\quad (n, k) = 1;\\ 0 & \text{if}\quad (n, k) ≠ 1. \end{cases} \]

See also Character group.

References

[1] A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[2] S.A. Morris, "Pontryagin duality and the structure of locally compact Abelian groups" , London Math. Soc. Lecture Notes , 29 , Cambridge Univ. Press (1977) MR0442141 Zbl 0446.22006
[3] E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 1 , Springer (1963) MR0156915 Zbl 0115.10603
How to Cite This Entry:
Character of a group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Character_of_a_group&oldid=36566
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article