Let $X$ be any set. A $\textdef{permutation group}$ on $X$ is any subgroup of $\sym{X}$.

• $\sym{X}$ denotes the symmetric group on $X$.
• Alternately, a permutation group is a subset of $\sym{X}$ that contains the identity and is closed under composition and taking inverses.

definition.permutation_group.tex

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\begin{definition}
Let $X$ be any set. A \textdef{permutation group} on $X$ is any subgroup of $\sym{X}$; that is, it is a subset of $\sym{X}$ that contains the identity and is closed under composition and taking inverses.
\end{definition}

• Permutation group

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