Definition:Symmetric Group/n Letters

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Let $S_n$ denote the set of permutations on $n$ letters.

Let $\struct {S_n, \circ}$ denote the symmetric group on $S_n$.

Then $\struct {S_n, \circ}$ is referred to as the symmetric group on $n$ letters.

It is often denoted, when the context is clear, without the operator: $S_n$.

Also known as

Some sources refer to this as the full symmetric group on $n$ letters.

Some sources use the term symmetric group of degree $n$.

Some sources use $\map S n$ or $\operatorname {Sym} \paren n$ for $S_n$.

Others use $\SS_n$ or some such variant.

Some older sources denote the symmetric group on $n$ letters as $\mathfrak S_n$.

However, this fraktur font is rarely used nowadays as it is cumbersome to reproduce and awkward to read.


In recognition that Symmetric Groups of Same Order are Isomorphic, it is unimportant to distinguish rigorously between symmetric groups on different sets.

Hence a representative set $S_n = \set {1, 2, \ldots, n}$ of cardinality $n$ is selected, and the symmetric group is defined on $S_n$ as the $n$th symmetric group.

As a consequence, results can be proved about the symmetric group on $n$ letters which then apply to all symmetric groups on sets with $n$ elements.

Also see

  • Results about the symmetric groups can be found here.