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If $G$ is a group acting on a set $S$, then the "orbit" of a point $x$ in $S$ is defined as the set of all elements of the form $gx$ where $g \in G$. My question: why was the word "orbit" chosen for this concept? It is not obviously related to previous uses of this word, such as the path of a planet around the Sun.

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1. Personally I do think the "orbit" in group theory carries the usual literal meaning of orbit.. 2. Source of the word: "ORBIT (in group theory). In a message posted to sci.math in 2009, Robert Israel wrote that Using MathSciNet the earliest reference he found to orbit that did not appear to be in the context of mechanics was Dick Wick Hall, “An example in the theory of pointwise periodic homeomorphisms,” Bull. Amer. Math. Soc. 45 (1939), Part 1:882-885: “For each point x of M the finite subset of M consisting of all the images of x under T is called the orbit of x under T.” – Soarer Jan 29 '11 at 6:40
in – Soarer Jan 29 '11 at 6:40
A somewhat similar question, though more general, was posted earlier today. The answers might be of interest… – Fredrik Meyer Jan 29 '11 at 7:11
The circle group acts by rotation on the set of possible locations of a planet with respect to the sun. An orbit of this group action is (up to the distinction between circles and ellipses) a planetary orbit in the usual sense. – Qiaochu Yuan Jan 29 '11 at 9:10
I always thought that the name "orbit" comes from the rotations of the plane as in Qiaochu Yuan's comment. Let me add that the Latin etymology leads back to "orbita", which is the eye socket (of a vaguely elliptical shape). – Andrea Mori Jan 29 '11 at 13:23
up vote 11 down vote accepted

It seems the concept goes back further than D.W. Hall (a student of Whyburn) - at least to Kuratowski, cf. the following excerpt from Kuczma et al. Iterative functional equations, p. 14. $\quad $ One can probably find further historical details by Googling "Kuratowski-Whyburn orbit" etc.

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Note: for a simple yet enlightening example of the key role that orbits play in the solution of functional equation see this post.

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