isotropy group


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isotropy group

[′ī·sə‚trō·pē ‚grüp]
(mathematics)
For an operation of a group G on a set S, the isotropy group of an element s of S is the set of elements g in G such that gs = s.
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For every e [member of] E we obtain a local representation [[rho].sub.e] : [G.sub.p(e)] [right arrow] P determined by [g.sup.-1] x e = e x [[rho].sub.e] (g) for g [member of] [G.sub.p(e)] where [G.sub.p(e)] is the isotropy group of p(e) [member of] X.
* The projection P [right arrow] P/[P.sub.[alpha]] has a local cross section where [P.sub.[alpha]] = {p [member of] P | p[alpha][p.sup.-1] = [alpha]} is the isotropy group of a under the conjugation action of P.
a (G x P)-CW-complex whose isotropy groups belong to F(R) and for which the K(H, a)-fixed point set [E.sub.F(R)] [(G x P).sup.K(H,[alpha])] is nonempty and weakly contractible for every (H, [alpha]) [member of] R.
A Fiber Bundle over [D.sub.3] = = SO(3, 2)/SO(3) x SO(2) whose H = SO(3) ~ SU(2) subgroup of the isotropy group (at the origin) K = SO(3) x x SO(2) acts on [S.sup.2] by simple rotations.
The base space is the 6D domain [B.sub.6] = SU(4)/U(3) = = SU(4)/SU(3) x U(1) whose subgroup SU(3) of the isotropy group (at the origin) K = SU(3) x U(1) acts on the internal symmetry space [CP.sup.2] via isometries.
The subgroup H = SO(5) of the isotropy group K = SO(5) x SO(2) acts on the Fibers F = [S.sup.4] (the internal symmetry space).
The subgroup H = SO(2) ~ U(1) of the isotropy group K = SO(2) x SO(2) acts on the fibers identified with the symmetry space [S.sup.1] (where the U(1) group acts).
One needs a Fiber Bundle over [D.sub.3] = = SO(3, 2)/SO(3) x SO(2) whose subgroup H = SO(3) of the isotropy group K = SO(3) x SO(2) acts on the internal symmetry space [S.sup.2] (the fibers).