Wikipedia says
'The difference between left and right actions is in the order in which a product like $gh$ acts on $x$. For a left action $h$ acts first and is followed by $g$, while for a right action $g$ acts first and is followed by $h$. Because of the formula $(gh)^{-1} = h^{-1}g^{-1}$, one can construct a left action from a right action by composing with the inverse operation of the group.'
I am struggling with the part where it says 'Because of the formula $(gh)^{-1} = h^{-1}g^{-1}$, one can construct a left action from a right action by composing with the inverse operation of the group.''
How can you construct left action from right action? if right action is $(g,x) \rightarrow xg$
any help will be appreciated
Thanks
Best Answer
If you didn't invert and you tried to turn $x \mapsto xg$ into a left-action then you'd get into trouble, because (if say $g \star x = xg$) then you'd have $$g \star (h \star x) = (xh)g = x(hg) = (hg) \star x \ne (gh) \star x$$ however if you had $g \star x = xg^{-1}$ then you'd have $$g \star (h \star x) = (xh^{-1})g^{-1} = x(h^{-1}g^{-1}) = x(gh)^{-1} = (gh) \star x$$ so everything's dandy.
Summary:
If $x \mapsto gx$ is a left action then $x \mapsto xg^{-1}$ is a right action.
If $x \mapsto xg$ is a right action then $x \mapsto g^{-1}x$ is a left action.
The idea is that inversion reverses the order of composition, so gives you right actions from left action and left actions from right actions.