# 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):

https://www.google.com/url?sa=t&source=web&rct=j&url=http://www.math.ku.dk/~olsson/manus/GruFus/Kurzweil-Stellmacher_Theory%2520of%2520finite%2520groups.pdf&ved=0ahUKEwiYmsTnrODPAhVB5SYKHcXHAMcQFggbMAA&usg=AFQjCNH-EJPLVWDku5ZBY2zwQqlEGCsz3g&sig2=MuavjBUWoHaq38F-F17FtA

• 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
• @paulgarrett I agree but this is supposed to be a well-known text in group theory so I'm sure someone can make sense of it given the theorem. I hope atleast! But thanks for verifying I wasn't just forgetting some special notation on my part. – Red Oct 16 '16 at 22:32
• 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
• @AndreasBlass Yes I understand negative one as the inverse notation, the question is really just about it being a superscript – Red Oct 16 '16 at 22:45
• This use of superscripts is explained in a parenthetical comment on page 10, which I admit is not the best place to explain a notation that's used on page 5. – Andreas Blass Oct 16 '16 at 22:56

## 1 Answer

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 $\phi(x) = ux$".