A $\textdef{permutation group}$ on $X$ is a set of permutations of $X$ that contains the identity permutation and is closed under function composition and taking inverses.

• Alternately, a permutation group is a subgroup of $\sym{X}$, where $\sym{X}$ denotes the symmetric group on $X$.

definition.permutation_group.tex

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\begin{definition}
A \textdef{permutation group} on $X$ is a set of permutations of $X$ that contains the identity permutation and is closed under function composition and taking inverses.
\end{definition}

• Permutation group

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