Let $(G, \cdot)$ be a group. Define $(G, *)$ as a group with the same underlying set and an operation $$a * b := b \cdot a.$$ What do you call such a group? What is the usual notation for it? I tried searching for 'dual group' and 'opposite group' with no results. It seems that this group is all you need to dispose of having to explicitly define right action, but there is no mention of such a group in wiki's 'Group action'. Am I mistaken and it's not actually a group?
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$\begingroup$I'm getting this as a first result by googling for opposite group: PlanetMath: opposite group.
All you want to know seems to be answered there and $G^{\operatorname{op}}$ is the usual notation.
A detailed verification that $G^{\operatorname{op}}$ is a group can be found by following the second google hit: Opposite Group - ProofWiki
$\endgroup$ 1 $\begingroup$A group is a groupoid (meaning a category all of whose arrows are invertible) with one object. If $\star$ denotes the object, group elements correspond to arrows $\star\to \star$, and multiplication $gh$ corresponds to composition $\star\xrightarrow{h} \star\xrightarrow{g} \star$. The opposite category of this category is the opposite group of this group: domain and codomain are reversed, so the old $gh$ is the new $hg$.
\Edit: Qiaochu Yuan was faster than me.
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