OppositeGroup 2007-05-27 21:05:28 2007-06-01 01:23:21 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{pstricks} \usepackage{psfrag} \usepackage{graphicx} \usepackage{amsthm} \usepackage{xypic} 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. Also, 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 $\varphi(x)=x^{-1}$. More generally, any anti-automorphism $\psi \colon G \to G$ gives rise to a corresponding isomorphism $\psi' \colon G \to G^{\mathrm{op}}$ via $\psi'(g)=\psi(g)$, since $\psi'(g*h)=\psi(g*h)=\psi(h)*\psi(g)=\psi(g)*'\psi(h)=\psi'(g)*'\psi'(h)$. 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 \PMlinkescapetext{right}, then a left action of $G^{\mathrm{op}}$ on $X$ can be defined by $g^{\mathrm{op}}x=xg$. \PMlinkescapetext{Similar} constructions occur in opposite ring and opposite category.