I'm having difficulties understanding the definition of the right regular representation as it appears in Dummit & Foote's Abstract Algebra text. On page 132 it says
Let $\pi:G \to S_G$ be the left regular representation afforded by the action of $G$ on itself by left multiplication. For each $g \in G$ denote the permutation $\pi(g)$ by $\sigma_g$, so that $\sigma_g(x)=gx$ for all $x \in G$. Let $\lambda:G \to S_G$ be the permutation representation afforded by the corresponding right action of $G$ on itself, and for each $h \in G$ denote the permutation $\lambda(h)$ by $\tau_h$. Thus $\tau_h(x)=xh^{-1}$ for all $x \in G$ ($\lambda$ is called the right regular representation of $G$).
I can't make since of that definition. Earlier, on page 129 the authors explain exactly what is the right action corresponding to some given left action:
For arbitrary group actions it is an easy exercise to check that if we are given a left group action of $G$ on $A$ then the map $A \times G \to A$ defined by $a \cdot g=g^{-1} \cdot a$ is a right group action. Conversely, given a right group action of $G$ on $A$ we can form a left group action by $g \cdot a=a \cdot g^{-1}$. Call these pairs corresponding group actions.
If I try to find the right group action corresponding to $g \cdot a=ga$, I get $a \cdot g:=g^{-1} \cdot a=g^{-1}a$. Hence it seems to me that the definition should be $\tau_h(x)=h^{-1}x$ and not $xh^{-1}$.
Are there any flaws with my reasoning?
Thanks!
Best Answer
In general, if $G$ acts on a set $S$ from the right via $(x,g) \mapsto x \cdot g$, then we obtain an action of $G$ on $S$ from the left via $g \cdot x := x \cdot g^{-1}$.
Let $*$ be the group multiplication in $G$. Then $G$ acts on $|G|$ (the underlying set of $G$, don't confuse it with $G$) from the right via $x \cdot g := x * g$. It follows that we obtain a left action from $G$ on $|G|$ via $g \cdot x := x \cdot g^{-1} = x * g^{-1}$. And this corresponds to a homomorphism of groups $G \to S(|G|)$ mapping $g$ to the permutation $(x \mapsto x * g^{-1}$).