====== Automorphism Group ======
====Definition==== 
Let $X$ be a set possibly equipped with additional structure, e.g. $X$ is the vertex set of a graph or $X$ is the underlying set of a group. The permutations of $X$ "preserving the additional structure" is called the $\textdef{automorphism group}$ of the structure, denoted $\aut(X)$.


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==== Remarks ==== 
  * If $X$ is a set with no additional structure, then $\aut(X)$ is the full [[symmetric group]] on $X$.
  * If $\mathcal{G}$ is a graph, i.e. a set $V$ of vertices together with a [[wp>symmetric relation|symmetric binary relation]] $E$ defining the edges, then a permutation $f$ of $V$ is a member of $\aut(\mathcal{G})$ if and only if whenever $x$ and $y$ are related by $E$ then $f(X)$ and $f(y)$ are related by $E$.


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==== $\LaTeX$ version ====
<file tex definition.automorphism_group.tex>
%%%%%
% DEPENDENCIES 
% RequiredMacros: \newcommand{\textdef}[1]{\textit{#1}}  \DeclareMathOperator{\aut}{Aut} 
%%%%%
\begin{definition}
Let $X$ be a set possibly equipped with additional structure, e.g. $X$ is the vertex set of a graph or $X$ is the underlying set of a group. The permutations of $X$ ``preserving the additional structure'' is called the \textdef{automorphism group} of the structure, denoted $\aut(X)$.
\end{definition}
</file>

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==== External links ====
  * [[wp>automorphism#Automorphism group|Automorphism group]]

==== Questions ====
> The words "preserving the additional structure" will be hard to have a student check. I would suggest that these words be clarified.  You did this in the remarks with $\aut(G)$.  Would it be better to change this page to Automorphism Group of a Graph. Then we could create Automorphism Group of a Set, and Automorphism Group of a Group.  
>> Type answers after >>.  If you remove rben from the tags once you have given an answer, then I'll look at it again.
{{tag>definition needsreview rben}}