# inverse of inverse in a group

Let $(G,*)$ be a group. We aim to prove that ${({a}^{-1})}^{-1}=a$ for every $a\in G$. That is, the inverse of the inverse of a group element is the element itself.

By definition $a*{a}^{-1}={a}^{-1}*a=e$, where $e$ is the identity in $G$. Reinterpreting this equation we can read it as saying that $a$ is the inverse of ${a}^{-1}$.

In fact, consider $b={a}^{-1}$, the equation can be written $a*b=b*a=e$ and thus $a$ is the inverse of $b={a}^{-1}$.

Title | inverse of inverse in a group |
---|---|

Canonical name | InverseOfInverseInAGroup |

Date of creation | 2013-03-22 15:43:36 |

Last modified on | 2013-03-22 15:43:36 |

Owner | cvalente (11260) |

Last modified by | cvalente (11260) |

Numerical id | 7 |

Author | cvalente (11260) |

Entry type | Proof |

Classification | msc 20-00 |

Classification | msc 20A05 |

Classification | msc 08A99 |

Related topic | AdditiveInverseOfAnInverseElement |