|
|
|
Revision difference : opposite group |
| Version 5 |
Version 4 |
| Let $G$ be a group under the operation $*$. The \emph{opposite group} of $G$, denoted $G^{\mathrm{op}}$, has the same underlying set as $G$, and its group operation is $*'$ defined by $g_1*'g_2=g_2*g_1$. |
Let $G$ be a group under the operation $*$. The \emph{opposite group} of $G$, denoted $G^{\mathrm{op}}$, has the same underlying set as $G$, and its group operation is $*'$ defined by $g_1*'g_2=g_2*g_1$. |
|
|
|
If $G$ is abelian, then it is equal to its opposite group. Moreover, every group $G$ (not necessarily abelian) is isomorphic to its opposite group: The \PMlinkname{isomorphism}{GroupIsomorphism} $\varphi \colon G \to G^{\mathrm{op}}$ is given by $\varphi(x)=x^{-1}$.
|
If $G$ is abelian, then it is equal to its opposite group. On the other hand, a nonabelian group $G$ can be isomorphic to its opposite group. For example, since ${S_3}^{\mathrm{op}}$ is a nonabelian group of order six, $S_3 \cong {S_3}^{\mathrm{op}}$.
|
|
|
| \PMlinkescapetext{Similar} constructions occur in opposite ring and opposite category. |
\PMlinkescapetext{Similar} constructions occur in opposite ring and opposite category. |
|
|
|
|
|
|
|
|
