If there is a function $a$ such that $a\circ g\circ a^{-1}=h$ then the functions $g$ and $h$ are conjugate to each other.
- If one wished to identify $a$, would one say "$g$ and $h$ are conjugate "by $a$" or "via $a$" or the like? What locution is most customary here? (Yes, one can write $a\circ g\circ a^{-1}=h$ as above, but some contexts make a more verbal style appropriate.)
- Is there a standard short name for the relation between $g$ and $h$ that for some involution $a$ (i.e. $a=a^{-1}$), $a\circ g\circ a = h$? Can one say "$g$ and $h$ are blahblah" or "$g$ is blahblah to $h$", where "blahblah" is something terse, like the word "conjugate"?
- Is there a standard name for a function that is conjugate to its own inverse?
- Is there a standard name for a function that is conjugate to its own inverse via an involution?
- If the answer to the foregoing is affirmative, one may wish to identify which involution it is in a verbal statement, e.g. "$g$ is blahblahblah via $a$". Is there some conventional verbiage for that?
(I've added a "category theory" tag because I think those writing about that topic may be the people with the most frequent occasion to use this kind of language.)
In a group, an element that's conjugate to its inverse is called a 'real' element (see here) and one conjugate to its inverse via an involution is called 'strongly real' (see here).