Suppose $A$ is some algebraic structure, and $x$ and $y$ are two elements of the underlying set. Is there any more concise way of stating, "there exists an automorphism in $A$ which maps $x$ to $y$"?. It seems like the obvious thing to say is "$x$ and $y$ are symmetric in $A$", but I've never come across this phrasing or any other in the literature, which is amazing to me since it's such an enormously useful concept.
Algebra terminology question
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terminology
universal-algebra
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3You could say x and y are in the same orbit of Aut(A), assuming the algebraic structure on A is understood, but that could be awkward. I don't think "symmetric" is a great term here. Unless you need to describe this concept 10 times or something, just say it the way you already did: there's an automorphism f such that f(x) = y. – 2010-08-17
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0"automorphically conjugate"? :) – 2010-08-17
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Some authors write that $x$ and $y$ are conjugate under $G$, or $G$-conjugate, e.g. from Bourbaki, Algebra:
Definition 5. Let $G$ be a group, $E$ a $G$-set and $x \in G$. An element $y \in E$ is conjugate to $x$ under the operation of $G$ if there exists an element $\alpha \in G$ such that $y = \alpha x$. The set of conjugate elements of $x$ is called the orbit of $x$ in $E$.
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0Tried to edit, but I think it should be $x \in E$, not $x \in G$. – 2012-12-27