I see many times the words "conjugacy" and "conjugation", and I don't really get the difference between the two. Especially, when we take an element of a group and want to say that this has some property "up to conjugacy/tion", which one is better, and why?

closed as too localized by Simon Thomas, Steven Landsburg, plusepsilon.de, Dan Petersen, Andres Caicedo May 16 '12 at 19:36
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To expand upon Gerhard's comment, I would add the following: In English, the acy suffix tends to denote the nounification of an adjective and the ation suffix tends to denote the nounification of a verb. Obviously, 'conjugate' can act as both an adjective and a verb e.g. '$x$ and $y$ are conjugate' or 'One may conjugate $x$ by some element to get $y$' Since these phrases are mathematically the same, there is thus no mathematical difference between '$x$ and $y$ are equal up to conjugacy' and '$x$ and $y$ are equal up to conjugation' 

