# Symmetric Group is Group

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## Theorem

Let $S$ be a set.

Let $\map \Gamma S$ denote the set of all permutations on $S$.

Then $\struct {\map \Gamma S, \circ}$, the symmetric group on $S$, forms a group.

## Proof 1

Taking the group axioms in turn:

### Group Axiom $\text G 0$: Closure

By Composite of Permutations is Permutation, $S$ is itself a permutation on $S$.

Thus $\struct {\map \Gamma S, \circ}$ is closed.

$\Box$

### Group Axiom $\text G 1$: Associativity

From Set of all Self-Maps under Composition forms Monoid, we have that $\struct {\map \Gamma S, \circ}$ is associative.

$\Box$

### Group Axiom $\text G 2$: Existence of Identity Element

From Set of all Self-Maps under Composition forms Monoid, we have that $\struct {\map \Gamma S, \circ}$ has an identity, that is, the identity mapping.

$\Box$

### Group Axiom $\text G 3$: Existence of Inverse Element

By Inverse of Permutation is Permutation, if $f$ is a permutation of $S$, then so is its inverse $f^{-1}$.

$\Box$

Thus all the group axioms have been fulfilled, and the result follows.

$\blacksquare$

## Proof 2

A direct application of Set of Invertible Mappings forms Symmetric Group.

$\blacksquare$