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G-Set


Let G be a group and S be a set. Then S is called a left G-set if there exists a map phi:G×S->S such that

 phi(g_1,phi(g_2,s))=phi(g_1g_2,s)

for all s in S and all g_1,g_2 in G. This is commonly written phi(g,s)=gs, so the above relation becomes

 g_1(g_2s)=(g_1g_2)s.

The map phi is called a left G-action on the set S.

Right G-sets and right G-actions are defined analogously except elements of G are multiplied by elements of S to the right instead of to the left. Left G-sets and right G-sets are both called G-sets for simplicity.

A G-set is an example of a group set, where G is the group in question.


See also

Group Set

This entry contributed by David Terr

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Cite this as:

Terr, David. "G-Set." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/G-Set.html

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