"conjugate to/with" or "conjugated to/with", a terminology question in group theory. This is a terminology question from a non-native English speaker.  Let $G$ be a group and $a,b\in G$ such that there exists $c\in G$ verifying :
$$b=cac^{-1} $$
I could say :


*

*the element $a$ is conjugate to $b$.

*the element $a$ is conjugate with $b$.

*the element $a$ is conjugated to $b$.

*the element $a$ is conjugated with $b$.

*the elements $a$ and $b$ are conjugate.

*the elements $a$ and $b$ are conjugated.

*the elements $a$ and $b$ are conjugate to each other.

*the elements $a$ and $b$ are conjugated to each other.
I really cannot figure out which one(s) is (are) correct, I think I have read them all at least one time in a paper or a course (maybe they are all correct, although $7$ and $8$ seem wrong...).
 A: Using the participle form "conjugated" is not necessary, I would avoid it. The proposition "with" doesn't really add anything, I would stick with "to" (or stick to "to"; prepositions are such fuzzy things). The general stylistic principle here is: longer words that add neither meaning nor intuition should be avoided. 
That leaves 1, 5, and 7, all of which are in common use and have slightly different intuition. In different contexts, one might choose a different one of 1, 5, or 7. Choice # 1 is good in an asymmetric context where you have already introduced $a$ and now want to point out some additional information, namely "$a$ is conjugate to $b$". Choices 5 and 7 would be better when $a$ and $b$ are being treated symmetrically. Choice $7$ might be better when $a,b$ are two elements of a big crowd.
A: I think this might help you:
https://proofwiki.org/wiki/Definition:Conjugate_(Group_Theory)
Using the example you've written, $a$ and $b$ are conjugate with respect to $c$ so all of your statements are equivalent except for number $6$ I would say which doesn't really make sense
