Let $X$ be any set. The $\textdef{symmetric group}$ on $X$, denoted $\sym(X)$, is the set of all permutations of $X$; that is, $\sym(X)$ is the set of all bijections from $X$ to $X$. We denote by $S_n$ the symmetric group on $X = \{1,2,\ldots, n\}$.

• None.

definition.symmetric_group.tex

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\begin{definition}
Let $X$ be any set. The \textdef{symmetric group} on $X$, denoted $\sym(X)$, is the set of all permutations of $X$; that is, $\sym(X)$ is the set of all bijections from $X$ to $X$. We denote by $S_n$ the symmetric group on $X = \{1,2,\ldots, n\}$.
\end{definition}

• Symmetric group

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