Let be a set possibly equipped with additional structure, e.g. is the vertex set of a graph or is the underlying set of a group. The permutations of “preserving the additional structure” is called the of the structure, denoted .

• If is a set with no additional structure, then is the full symmetric group on .
• If is a graph, i.e. a set of vertices together with a symmetric binary relation defining the edges, then a permutation of is a member of if and only if whenever and are related by then and are related by .

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 . 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.