Fundamental Domain

Let G be a group and S be a topological G-set. Then a closed subset F of S is called a fundamental domain of G in S if S is the union of conjugates of F, i.e.,

 S= union _(g in G)gF,

and the intersection of any two conjugates has no interior.

For example, a fundamental domain of the group of rotations by multiples of 180 degrees in R^2 is the upper half-plane {(x,y)|y>=0} and a fundamental domain of rotations by multiples of 90 degrees is the first quadrant {(x,y)|x,y>=0}.

The concept of a fundamental domain is a generalization of a minimal group block, since while the intersection of fundamental domains has empty interior, the intersection of minimal blocks is the empty set.

See also

G-Set, Group Block

Portions of this entry contributed by David Terr

Portions of this entry contributed by Richard Peterson

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Cite this as:

Peterson, Richard; Terr, David; and Weisstein, Eric W. "Fundamental Domain." From MathWorld--A Wolfram Web Resource.

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