Superscript Function Notation(variable as base) $x^{\phi ^{-1}}$

I apologize if this is a silly question but I cannot find the necessary information elsewhere on the internet: If $\phi : u \mapsto ux$ is a function from $U$ to $U$ , does the notation $x^{\phi ^{-1}}$ mean the inverse function of $\phi$ where the domain variable is $x$??

For full context of the notation Ill put a link to Kurzweil-Theory book on groups where it is on page 5 (page 18 of pdf):

• The context is already confusing: it is not current practice to say that a function from a set $U$ to itself would be written as $\phi:u\Rightarrow ux$... And I've never seen the notation $x^{\ph^{-1}$, and would not be able to easily guess its intent. There's no universal or widely-understood notion of "domain variable", either. Something's wrong here... – paul garrett Oct 16 '16 at 22:24
• The source of the confusion seems to be that you miscopied a symbol. It should be $\phi:u\mapsto ux$. So $\phi$ is the function $U\to U$ defined by $\phi(u)=ux$ for all $u\in U$. And $x^{\phi^{-1}}$ means what I'd usually write as $\phi^{-1}(x)$. – Andreas Blass Oct 16 '16 at 22:39
It is not uncommon in group theory to write $x^g$ for $g^{-1}xg$: i.e., what you get if you apply conjugation by $g$ to $x$, where $x$ and $g$ are members of some group $G$ (this can save quite a lot of ink). The conjugation function $x \mapsto x^g$ is called an inner automorphism on $G$. By extension of this notational idea, it isn't uncommon to write $x^{\alpha}$ for the result $\alpha(x)$ of applying an arbitrary automorphism $\alpha$ on $G$ to $x$.
The authors of this book have clearly taken this further and write $x^{f}$ for $f(x)$ for functions $f$ in general, as in in your quotation (where $f$ is $\phi^{-1}$). The notation $\phi : x \mapsto ux$ is a fairly standard way of saying "the function $\phi$ defined by $\phi(x) = ux$".