Isotropy Group

Some elements of a group G acting on a space X may fix a point x. These group elements form a subgroup called the isotropy group, defined by

 G_x={g in G:gx=x}.

For example, consider the group SO(3) of all rotations of a sphere S^2. Let x be the north pole (0,0,1). Then a rotation which does not change x must turn about the usual axis, leaving the north pole and the south pole fixed. These rotations correspond to the action of the circle group S^1 on the equator.

When two points x and y are on the same group orbit, say y=gx, then the isotropy groups are conjugate subgroups. More precisely, G_y=gG_xg^(-1). In fact, any subgroup conjugate to G_x occurs as an isotropy group G_y to some point y on the same orbit as x.

See also

Effective Action, Free Action, Group Action, Group Orbit, Group Representation, Lie Group Quotient Space, Matrix Group, Topological Group, Transitive

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Isotropy Group." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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