# opposite group

Let $G$ be a group under the operation^{} $*$. The *opposite group* of $G$, denoted ${G}^{\mathrm{op}}$, has the same underlying set as $G$, and its group operation^{} is ${*}^{\prime}$ defined by ${g}_{1}{*}^{\prime}{g}_{2}={g}_{2}*{g}_{1}$.

If $G$ is abelian^{}, then it is equal to its opposite group. Also, every group $G$ (not necessarily abelian) is isomorphic^{} to its opposite group: The isomorphism^{} (http://planetmath.org/GroupIsomorphism) $\phi :G\to {G}^{\mathrm{op}}$ is given by $\phi (x)={x}^{-1}$. More generally, any anti-automorphism $\psi :G\to G$ gives rise to a corresponding isomorphism ${\psi}^{\prime}:G\to {G}^{\mathrm{op}}$ via ${\psi}^{\prime}(g)=\psi (g)$, since ${\psi}^{\prime}(g*h)=\psi (g*h)=\psi (h)*\psi (g)=\psi (g){*}^{\prime}\psi (h)={\psi}^{\prime}(g){*}^{\prime}{\psi}^{\prime}(h)$.

Opposite groups are useful for converting a right action to a left action and vice versa. For example, if $G$ is a group that acts on $X$ on the , then a left action of ${G}^{\mathrm{op}}$ on $X$ can be defined by ${g}^{\mathrm{op}}x=xg$.

Title | opposite group |
---|---|

Canonical name | OppositeGroup |

Date of creation | 2013-03-22 17:09:56 |

Last modified on | 2013-03-22 17:09:56 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 10 |

Author | Wkbj79 (1863) |

Entry type | Definition |

Classification | msc 08A99 |

Classification | msc 20-00 |