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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?

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closed as too localized by Simon Thomas, Steven Landsburg, Marc Palm, Dan Petersen, Andrés E. Caicedo May 16 '12 at 19:36

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I have never seen any context in which these two phrases have different meanings. – Neil Strickland May 16 '12 at 17:13
I see the difference as one of perspective. Up to conjugacy for me suggests a perspective where we are above looking down at a collection of classes, and talking about a member in one conjugacy class because we are in a position to shift easily between classes. Up to conjugation for me suggests we are down working with a particular example, perhaps we have a word we are using, and what we are interested holds up to conjugation, but then we have to rewrite the word ("climb above") and work to show the desired property literrally. Gerhard "Sees From Many Sides Now" Paseman, 2012.05.16 – Gerhard Paseman May 16 '12 at 17:19

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'

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Thinking to the Latin and Greek origin, I'd also say add that the -tion suffix forms verbal nouns denoting an action (derived from the verb) while the suffix -cy mostly denotes a quality or a condition (from the adjective). It seems that this fits with Gerhard explanation. – Pietro Majer May 17 '12 at 7:56
(So, while there is no mathematical difference between '$x$ and $y$ are equal up to conjugacy' and '$x$ and $y$ are equal up to conjugation' the former sentence would just mean that $x$ and $y$ are in the same conjugacy class, the latter should suggest that you can transform $x$ into $y$ operating a conjugation on it). – Pietro Majer May 17 '12 at 10:47
Thanks for all your comments. I especially like the last comment of Pietro. – Jérémy Blanc May 17 '12 at 11:00

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