Somebody know this notation in group theory:


where $G$ is a group and $X$ aparently is a subset of G?

I've come across with this notations in the following problem:

Show that $X^G = \langle \{ gxg^{-1};\; g\in G,\; x \in X \} \rangle.$


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    $\begingroup$ How do you define $X^G$? Sometimes that is the definition. $\endgroup$ – Pedro Tamaroff Sep 17 '13 at 2:04
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    $\begingroup$ It is moderately common notation for the conjugacy class of $X$. $\endgroup$ – André Nicolas Sep 17 '13 at 2:18
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    $\begingroup$ This notation is also used to denote the set of fixed points for the action of G on X. I expect the book you are reading to define this notation before the exercise. Also, look at the index in the end if the book. $\endgroup$ – Moishe Kohan Sep 17 '13 at 3:03
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    $\begingroup$ A symbol with a superscript is too useful a notation not to be used in several, context dependent ways. The authors of textbooks know this, so they include a definition somewhere. Here it looks like it might mean the union of orbits of a group $G$ acting on a set, and in this specific exercise $G$ is acting on itself by conjugation, and $X$ is a subset (making the exercise a tautology). I would think that it is more common (YMMV) to denote the set of fixed points of the action in this way, but that doesn't fit the claim in the exercise. $\endgroup$ – Jyrki Lahtonen Sep 17 '13 at 3:49
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    $\begingroup$ The main point is the THERE IS NO UNIQUE UNIVERSAL MEANING TO THIS NOTATION. Anyone not reading the same book can only make educated guesses. Why did you think differently? $\endgroup$ – Jyrki Lahtonen Sep 17 '13 at 3:50

In group theory, this notation is normally used to define the normal closure in $G$ of the subset $X$. This is the smallest normal subgroup of $G$ containing $X$. With that definition you can easily prove that $X^G = <g^{-1}xg | g \in G, x\in X>$. It also equals the intersection of all normal subgroups of $G$ containing $X$.

Note that in other parts of mathematics the notion of closure is very common and defined in a similar way. For example in topology one has normal closures of sets, and in field theory, you will encounter algebraic closures.

  • $\begingroup$ I have only rarely seen this for the normal closure, and not in recent papers (since, like, pre-$\TeX$, when they wrote them in by hand!). More common is $\langle\langle X\rangle\rangle$. $\endgroup$ – user1729 Sep 17 '13 at 12:51
  • $\begingroup$ If you consider Marty Isaacs as an authority (I do!) then consult his book Finite Group Theory on page 51 where the definition of normal closure is given, together with the notation as indicated. This is just one of the many places where this is used. Again, the noation is very common in group theory. And, as far as I am concerned, $<<X>> =<X>$, so double brackets would be very strange. $\endgroup$ – Nicky Hekster Sep 17 '13 at 18:26
  • $\begingroup$ Perhaps it is just the different worlds of group theory. I work in geometric and combinatorial group theory where actions are all important and $X^G$ is often used to denote the fixed point of an action. I suppose the moral of this story is that notation should always come with a definition, be it only a two words! $\endgroup$ – user1729 Sep 17 '13 at 18:45
  • $\begingroup$ Check out the books of Lyndon & Schupp or Magnus, Karras & Solitar (both on combinatorial group theory), I bet they have the notation. D.J.S. Robinson's book "A Course in the Theory of Groups" denotes it in the same way. $\endgroup$ – Nicky Hekster Sep 17 '13 at 21:25
  • $\begingroup$ Yeah, I picked up on Robinson's book which is why I wrote "geometric and combinatorial" as opposed to just infinite ;-) . Magnus, Karrass and Solitar don't use any notation for it, so far as I can tell! And Lyndon and Schupp is in my office...(although definately the notation $\langle\langle R\rangle\rangle$ is quite common.) $\endgroup$ – user1729 Sep 17 '13 at 21:30

My understanding:

Let $G$ act on a set $A$ and $B$ be a subset of $A$. Then one usually denotes $GB=\{gb| g\in G, b\in B\}$. The author used $X^G$ instead of $GX$ ($G$ acts on $G$, $X\subset G$) because some people write $x^g$ instead of $gx$.

The standard meaning of $X^G$ is in the remark of studiosus (the set of fixed points).

  • $\begingroup$ I disagree that this is standard. It is common, yes, but I don't think you could call it standard... $\endgroup$ – user1729 Sep 17 '13 at 12:52
  • $\begingroup$ @user1729: OK, maybe. I don't know English well. $\endgroup$ – Boris Novikov Sep 17 '13 at 13:16

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