# a characterization of groups

###### Theorem.

A non-empty semigroup^{} $S$ is a group
if and only if
for every $x\mathrm{\in}S$ there is a unique $y\mathrm{\in}S$ such that $x\mathit{}y\mathit{}x\mathrm{=}x$.

###### Proof.

Suppose that $S$ is a non-empty semigroup, and for every $x\in S$ there is a unique $y\in S$ such that $xyx=x$. For each $x\in S$, let ${x}^{\prime}$ denote the unique element of $S$ such that $x{x}^{\prime}x=x$. Note that $x({x}^{\prime}x{x}^{\prime})x=(x{x}^{\prime}x){x}^{\prime}x=x{x}^{\prime}x=x$, so, by uniqueness, ${x}^{\prime}x{x}^{\prime}={x}^{\prime}$, and therefore ${x}^{\prime \prime}=x$.

For any $x\in S$, the element $x{x}^{\prime}$ is idempotent^{} (http://planetmath.org/Idempotency),
because ${(x{x}^{\prime})}^{2}=(x{x}^{\prime}x){x}^{\prime}=x{x}^{\prime}$.
As $S$ is nonempty, this means that $S$ has at least one idempotent element.
If $i\in S$ is idempotent,
then $ix=ix{(ix)}^{\prime}ix=ix{(ix)}^{\prime}iix$, and so ${(ix)}^{\prime}i={(ix)}^{\prime}$,
and therefore ${(ix)}^{\prime}={(ix)}^{\prime}{(ix)}^{\prime \prime}{(ix)}^{\prime}={(ix)}^{\prime}ix{(ix)}^{\prime}={(ix)}^{\prime}x{(ix)}^{\prime}$,
which means that $ix={(ix)}^{\prime \prime}=x$.
So every idempotent is a left identity^{},
and, by a symmetric^{} argument, a right identity.
Therefore, $S$ has at most one idempotent element.
Combined with the previous result,
this means that $S$ has exactly one idempotent element,
which we will denote by $e$.
We have shown that $e$ is an identity^{},
and that $x{x}^{\prime}=e$ for each $x\in S$, so $S$ is a group.

Conversely, if $S$ is a group then $xyx=x$ clearly has a unique solution, namely $y={x}^{-1}$. ∎

Note. Note that inverse semigroups do not in general
satisfy the hypothesis^{} of this theorem:
in an inverse semigroup there is for each $x$ a unique $y$ such that $xyx=x$ and $yxy=y$,
but this $y$ need not be unique as a solution of $xyx=x$ alone.

Title | a characterization of groups |
---|---|

Canonical name | ACharacterizationOfGroups |

Date of creation | 2013-03-22 14:45:08 |

Last modified on | 2013-03-22 14:45:08 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 10 |

Author | yark (2760) |

Entry type | Theorem |

Classification | msc 20A05 |

Related topic | Group |

Related topic | RegularSemigroup |

Related topic | AlternativeDefinitionOfGroup |