I am just starting to learn about groups, and the concept of conjugation came up. I was wondering what the relationship was, if any, between conjugation in the group sense and conjugation in the algebraic sense. And, why is it called conjugation?
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1$\begingroup$ what do you mean with algebraic sense? $\endgroup$– Daniel ValenzuelaSep 3, 2014 at 7:29
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$\begingroup$ Sorry, like the term $(a-b)*(a+b)$. $\endgroup$– user170141Sep 3, 2014 at 18:14
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2 Answers
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The idea of conjugates (in the context of simple radical expressions, used in order to clear denominators of radicals) generalizes in Galois theory. An algebraic element (something that satisfies a polynomial relation) has as its conjugates the other roots of its minimal polynomial, which as a whole make what's called the orbit of that element under the action of the Galois group. Speaking of a group alone, a group also acts on itself by conjugation, and the conjugates of an element are precisely its orbit under this action. However I don't think the word "conjugate" applies in an arbitrary group action settting, only when a group acts on itself by [what we call conjugation] and when a Galois group acts on algebraic elements in a field extension. So the lack of presence of the label outside these two specific examples of group actions, where they otherwise could be, indicates this is some measure of a historical accident.
The use of the word "conjugate" often implies the pairing of two things, like two sides of the same coin or two partners in a conjugal visit, which also suggests a symmetry among things that we call "conjugate" in daily life. This is why, etymologically, it makes some sense to use the word conjugate to describe elements that are related by symmetry in this way, even if it only turned out to be used in two examples of this context. One then calls the group action "conjugation" because it shuffles conjugates, probably.
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Not sure that this is an answer for your first question, but as to the last: the Oxford English Dictionary gives various meanings for the word, most relating to being "joined together", or "derived from the same root or stem". Compare also "conjugal", relating to marriage.
As is often the case, the mathematical usage of the word is distinctly metaphorical. But I think we can probably agree, if rather loosely, that
complex conjugates are "joined" by having the same real part;
algebraic conjugates are "derived from the same" minimal polynomial;
group conjugates are "derived from the same" normal subgroup.
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2$\begingroup$ I am not sure I understand what you mean by your last bullet point. $\endgroup$ Sep 3, 2014 at 8:01
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$\begingroup$ @GeoffRobinson $N$ is a normal subgroup of $G$ if for every $n$ in $N$, every conjugate $g^{-1}ng$ is also in $N$. Maybe this doesn't count as "derived from", but as I said, it's a rather loose usage. $\endgroup$– DavidSep 3, 2014 at 10:17
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1$\begingroup$ @David: I'd be surprised if Geoff doesn't know the definition of a normal subgroup. OK, his user profile doesn't give much hints, but you could take a look at the answers he gave here and at MO. $\endgroup$– j.p.Sep 3, 2014 at 15:26
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$\begingroup$ @j.p. I'd be surprised too, but I couldn't think what else he meant by his comment ;-) $\endgroup$– DavidSep 4, 2014 at 0:09
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$\begingroup$ @David: Even loosely, I cannot guess either what you intend to express with your last bullet point. Would you be so nice to expand this thought a bit? (Thanks!) $\endgroup$– j.p.Sep 4, 2014 at 10:41