Group Set

A group set is a set whose elements are acted on by a group. If the group G acts on the set S, then S is called a G-set.

Let G be a group and let S be a G-set. Then for every element s of S and every element g of G, an element gs of S is associated in such a way that es=s, where e is the identity element of G and such that (g_1g_2)s=g_1(g_2s) for every pair of elements g_1,g_2 in s.

The multiplication of elements of S by elements of G described above is called a left G-action, since the elements of G are multiplied on the left. For this reason, S is sometimes called a left G-set. Right G-sets are defined analogously to left G-sets except that multiplication by elements of G is now performed to the right of elements of S.

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This entry contributed by David Terr

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Terr, David. "Group Set." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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