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I am using conjugation frequently in a paper, and I don't know the standard notation. Here http://mathworld.wolfram.com/Conjugation.html they use the notation $\phi_x(g)=xgx^{-1}$, but here https://en.wikipedia.org/wiki/Adjoint_representation they use the notation $\psi_g(h)=ghg^{-1}$. Here https://groupprops.subwiki.org/wiki/Group_acts_as_automorphisms_by_conjugation they use the notation $c_g$. I know this is related to the adjoint representation of a Lie group, but I'm not sure that the adjoint representation is appropriate here. Any help would be appreciated.

J. W. Tanner
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Teddy Baker
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2 Answers2

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Group theorists often use $g^x$ as shorthand for $x^{-1}gx$. Then $$g^{xy}=y^{-1}x^{-1}gxy=y^{-1}g^xy=(g^x)^y$$ etc.

Angina Seng
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  • Thank you for the answer, I appreciate it, but I don't like that notation because I am using quaternions and that can easily be confused with the exponential. Is that the only notation that is standardized? – Teddy Baker May 02 '17 at 04:14
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    @Teddy: Beside this notation, and something you mentioned in question, there are no other notations I have seen in many papers on algebra. If you want to mix-up with exponential, then simply, you have to remember whether exponent is group element (which shows conjugation) or some number (which is usual power). Notation said by Lord is interesting because, for example, $(g^x)^{10}=(g^{10})^x$, and $(g^x)^{-1}=(g^{-1})^x$, .. – p Groups May 02 '17 at 04:56
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    The exponential notation is probably the most common, but the notation $c_g(x) = gxg^{-1}$ is used quite often in subjects like group cohomology or algebraic topology (when group actions come up). – Goa'uld May 02 '17 at 15:49
  • Ok great, maybe I will use the notation $c_g(x)$, given the issue I had with the exponential notation. Thanks! – Teddy Baker May 02 '17 at 20:18
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    It usually comes with the notation $^xg=xgx^{-1}$, which itself cannot be confused with exponents. – YCor May 02 '17 at 21:40
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    Its worth pointing out that $g^x$ can mean either $x^{-1}gx$ or $xgx^{-1}$, depending on the author (I presume it is the same for $c_g(x)$). So you should say which you mean when writing. – user1729 Dec 12 '18 at 12:53
  • @user1729: I disagree: $xgx^{-1}$ is usually denoted ${}^xg$. – Arturo Magidin Nov 11 '19 at 02:22
  • @Arturo I have never seen the notation $^xg$ before YCor's comment, and haven't seen it since. On the other hand, I am pretty sure I have seen $g^x$ written to mean $xgx^{-1}$ (but cannot find an example!). Even if this is the "standard" meaning for this motation it's still helpful to define notation like this, even if it's just for the sake of first-year pH.D. students... – user1729 Nov 11 '19 at 06:42
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    @user1729: The problem with defining $g^x$ as $xgx^{-1}$ is that the notation does not satisfy the obvious property you want it to satisfy by analogy: note that $g^{xy}$ would mean $(xy)g(xy)^{-1} = xygy^{-1}x^{-1}$, but that $(g^x)^y$ would mean $yxgx^{-1}y^{-1} = g^{yx}$. So you do not have $(g^x)^y = g^{xy}$. Whereas ${}^x({}^yg) = {}^{xy}g$, and $g^x = x^{-1}gx$ does satisfy $(g^x)^y = g^{xy}$. Using $g^x$ to denote $xgx^{-1}$ is bad notation, because that type of conjugation is a left action but not a right action. – Arturo Magidin Nov 11 '19 at 07:02
  • @user1729: P.S. I apologize for doing this so much later than original comment, as I did not notice the date at first, given the recent edit. I will look for an instance of the notation in some of my books tomorrow. – Arturo Magidin Nov 11 '19 at 07:03
  • @user1729: Some examples of use of that notation: Brown, Johnson, and Robertson, "Some computation of nonabelian tensor products of groups", J. Algebra 111 (1987) no 1, 177-202 MR 913203 (88m:20071). Brown and Loday, "Van Kampen theorems for diagram spaces", Topology 26 (1987) no 3, 311-335. Rotman's book uses the notation you propose (bad! bad Rotman!) though only briefly from what I can tell by browsing. – Arturo Magidin Nov 11 '19 at 15:29
  • @Arturo Glad Rotman uses that notation and I didn't just make it up! I understand your point about it being bad because it looks messy with group actions. However, I think writing $g^x$ for $xgx^{-1}$ is more for notational convenience (as $xgx^{-1}$ looks messy and is harder to parse) than for any mathematical benefit. [And don't worry about the dates - my previous comment was a year after the one before it!] – user1729 Nov 12 '19 at 21:14
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If you are concerned about possible confusion with exponentiation (given that, for example, $-1$ will be both a group element and an integer), and given that you are conugating on the left, one possibility is to switch to the "other" conjugation notation compatible with the left action you are using: $g\mapsto xgx^{-1}$. That conjugation is usually denoted by ${}^xg$. So then regular exponentiation acts on the right, and group conjugation acts on the left. Thus, "${}^xg^n$" means either of $(xgx^{-1})^n$, or $x(g^n)x^{-1}$ (which fortunately are equal).

I'm afraid that these are only standard notations for conjugation. The alternative is to use maps as you do.

Arturo Magidin
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