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

If ${\displaystyle G}$  is abelian, then it is equal to its opposite group. Also, every group ${\displaystyle G}$  (not necessarily abelian) is naturally isomorphic to its opposite group: An isomorphism ${\displaystyle \varphi :G\to G^{\mathrm {op} }}$  is given by ${\displaystyle \varphi (x)=x^{-1}}$ . More generally, any antiautomorphism ${\displaystyle \psi :G\to G}$  gives rise to a corresponding isomorphism ${\displaystyle \psi ':G\to G^{\mathrm {op} }}$  via ${\displaystyle \psi '(g)=\psi (g)}$ , since

${\displaystyle \psi '(g*h)=\psi (g*h)=\psi (h)*\psi (g)=\psi (g){\mathbin {\ast '}}\psi (h)=\psi '(g){\mathbin {\ast '}}\psi '(h).}$ 

Let ${\displaystyle X}$  be an object in some category, and ${\displaystyle \rho :G\to \mathrm {Aut} (X)}$  be a right action. Then ${\displaystyle \rho ^{\mathrm {op} }:G^{\mathrm {op} }\to \mathrm {Aut} (X)}$  is a left action defined by ${\displaystyle \rho ^{\mathrm {op} }(g)x=x\rho (g)}$ , or ${\displaystyle g^{\mathrm {op} }x=xg}$ .

• Opposite ring
• Opposite category

• http://planetmath.org/oppositegroup