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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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.
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