Definition:Symmetric Group

Definition

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

The structure $\left({S_n, \circ}\right)$, where $\circ$ denotes composition of mappings, forms a group.

This is called the symmetric group on $n$ letters, and is usually denoted, when the context is clear, without the operator: $S_n$.

From Symmetric Group is a Group, we have that $\left({S_n, \circ}\right)$ is isomorphic to the group of permutations of the $n$ elements of any set $T$ whose cardinality is $n$.

In order not to make notation overly cumbersome, the product notation is usually used for composition, thus $\pi \circ \rho$ is written $\pi \rho$.

Also, for the same reason, rather than using $I_{S_n}$ for the identity mapping, the symbol $e$ is usually used.

Also known as

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

Some sources use $S \left({n}\right)$ for $S_n$.

Others use $\mathcal S_n$ 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.

Also see

Some sources refer to the group of permutations of any general set $S$ with $n$ elements as the (full) symmetric group on $S$.

For finite groups it matters little, as (by this result) all such groups are isomorphic anyway.

It can of course be convenient sometimes to be able to refer unambiguously to the contents of this group by using cycle notation (or indeed, two-row notation if you really want to) without confusion. If all such definitions are based on an underlying set containing a rigorously specified set of natural numbers, this makes certain aspects of this discipline significantly easier.

We can stretch the definition for countable $S$, as in that case there is a bijection between $S$ and $\N$ by definition of countability.

However, this definition can not apply if $S$ is uncountable.

• Results about the symmetric groups can be found here.

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 4.5$: Example $79$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 7$: Example $7.5$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 1.6$: Theorem $5$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 30$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 34 \ (4)$