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 $\aut(X)$.

• 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 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 $E$.

definition.automorphism_group.tex

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% DEPENDENCIES 
% RequiredMacros: \newcommand{\textdef}[1]{\textit{#1}}  \DeclareMathOperator{\aut}{Aut} 
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\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}

• Automorphism group

• 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.

definition needsreview rben