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.

Let $X$ be any set. A permutation on $X$ is a bijection on $X$. A $\textdef{permutation group}$ on $X$ is a set of permutations on $X$ that forms a group, where we use function composition as multiplication.

• $\sym{X}$ denotes the symmetric group on $X$.
• The first definition requires that we first define symmetric group, subgroup, and that we have developed some intuition about how to show something is a subgroup. The second definition avoids this.

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

definition needsreview rben