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opposite group (Definition)

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 $ *'$ 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 isomorphism $ \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 right, then a left action of $ G^{\mathrm{op}}$ on $ X$ can be defined by $ g^{\mathrm{op}}x=xg$.

Similar constructions occur in opposite ring and opposite category.



"opposite group" is owned by Wkbj79.
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Cross-references: opposite category, opposite ring, occur in, acts on, left action, right action, isomorphism, anti-automorphism, isomorphic, abelian, group operation, operation, group
There is 1 reference to this entry.

This is version 7 of opposite group, born on 2007-05-27, modified 2007-06-01.
Object id is 9477, canonical name is OppositeGroup.
Accessed 845 times total.

Classification:
AMS MSC: 20-00 (Group theory and generalizations :: General reference works )
 08A99 (General algebraic systems :: Algebraic structures :: Miscellaneous)
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