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Definition of conjugates: Conjugate wiki link.

Suppose $G$ is a group. Two elements $a$ and $b$ of $G$ are called conjugate if there exists an element $g$ in $G$ with $g*a*g^{−1} = b$. Here $*$ is operation on group.

Question: If $g * g^{−1}$ will give us identity. Then equation will become $a * e = b$

hence $a = b$.

Then why we call them conjugate of each other?

I am missing basic thing here. Please help to understand conjugate.

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    $\begingroup$ Groups need not be commutative. $\endgroup$ Oct 17, 2016 at 10:08
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    $\begingroup$ The group need not be commutative $\endgroup$ Oct 17, 2016 at 10:08
  • $\begingroup$ @Tobias Kildetoft and @ G. Snapsmath: Thanks you ...! Perfect... This will not hold for non abelian groups. $\endgroup$
    – Omkara
    Oct 17, 2016 at 10:14
  • $\begingroup$ This blog post may be helpful: axiomtutor.com/new-blog/2022/9/28/… $\endgroup$
    – Addem
    Oct 5, 2022 at 22:55

2 Answers 2

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As some comments mentioned, conjugation is only really useful in non-abelian groups. Here are a few other things that may be useful to know:

  1. We say "conjugation by $u$" for the action of taking some element, $g$ say, to $u^{-1}gu.$ It is easy to see that this is an isomorphism (automorphism if you like).
  2. The relation "$a$ is conjugate to $b$" is an equivalence relation. We call the classes conjugacy classes.
  3. An intuition for conjugation is that $u^{-1}gu$ is looking at $g$ from the point of view of $u$. For example you may know how to solve some problem in some special case (e.g. The North Pole of a sphere or the point $\infty$ on the projective plane) and then you can use conjugation to solve the problem more generally (i.e. Conjugating by the element which moves your point of interest to the North Pole or $\infty$ in the vague examples I gave).
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  • $\begingroup$ Thanks for nice explanation. Could you please elaborate on point#3, how can can we use conjugation to solve problem more generally? Please provide reference to understand conjugation the way you explained. Similarly please provide reference links/books/videos-links to intuitively study Group Theory (abstract algebra). $\endgroup$
    – Omkara
    Oct 17, 2016 at 14:45
  • $\begingroup$ @Omkara I want to give a good example that you can understand? Which of these are you most familiar with? 1) Symmetric groups, 2) Möbius transforms (and the group of all Möbius transformations), 3) The UHP model for hyperbolic space (and the group of it's isometries), or 4) the isometries of 3D Euclidean space (i.e. The group of rotations, translations and reflections of 3D space) $\endgroup$ Oct 17, 2016 at 15:49
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    $\begingroup$ Im afraid I definitely don't know any videos and can't think of any books at the moment to help you. One normally doesn't tend to try to only learn the intuition but to build it by seeing the applications of a theory and understanding proofs in and of it. Often a book will have remarks about how to think intuitively about a concept however. $\endgroup$ Oct 17, 2016 at 15:52
  • $\begingroup$ @DanRobertson Hi! it is very helpful, I upvoted ur answer, also i wanted to have some examples for demonstrations, use symmetric group as it is easy to visualize in general, I am really interested in your third point (special to general), at least provide a symmetric group example for that, plz. $\endgroup$ Sep 10, 2021 at 17:54
  • $\begingroup$ @MRJames2017 Sorry, I have, in the last four years, forgotten whatever example I had in the symmetric group and I am no longer a maths student so I don’t think I can produce any good examples off the top of my head. A simple case where thinking about conjugation is useful is maybe something like trying to prove that $S_n$ is generated by e.g. $(12)$ and $(123\dots n)$ (not even sure if it is true though). $\endgroup$ Sep 14, 2021 at 12:55
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Conjugacy is one of the more important concepts in group theory.

It's an equivalence relation, an easy exercise.

The class equation gives, for a finite group, the sizes of the various conjugacy classes. This equation is very useful. For instance you can read off the size of the center (the sum of the ones, since elements of the center are in classes of size one), and often tell which subgroups are normal (unions of conjugacy classes).

For an abelian group we just get a string of ones.

Functions that are constant on conjugacy classes are called class functions. As an example there's characters. They were important in Feit-Thompson's theorem, leading to the classification of finite simple groups.

One of the most basic types of group action is conjugation by an element of the group.

The notion also comes up often when dealing with free groups.

Finally for now, conjugate elements in a group always have the same order (since conjugation by an element of the group is an automorphism). This comes in handy when working with group presentations, among other things. For instance when two elements are conjugate and have relatively prime order, they are the identity element.

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  • $\begingroup$ Burnside himself disproved in 1913 his conjecture (dated 1911) . $\endgroup$
    – citadel
    Feb 25 at 7:16
  • $\begingroup$ Right I remember that. I took that part out though. @citadel $\endgroup$
    – me too
    Feb 25 at 8:22

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