97j:22030 22E46

Greenlees, J. P. C. ^{(}4-SHEF-PS^{)};

Pérez, J. A. [Pérez, Juan Antonio]

Connected Lie groups that act freely on a product of linear spheres. (English. English summary) Bull. London Math. Soc. 28 (1996), no. 6, 634–642. The authors prove that a compact connected Lie group G admits a free action on a product of linear spheres if and only if G is a product of a torus and some SU(2)’s modulo some central elementary abelian 2-group which intersects the product of SU(2)’s trivially. To show that G cannot contain other simple factors, the authors observe that every compact simple group besides SU(2) contains a subgroup isomorphic to SO(3) or Spin(5). It is routine to show that SO(3) and Spin(5) have a line fixed by a one-dimensional torus in any finitedimensional representation V, and therefore a point in the sphere of V with nontrivial stabilizer. The condition on the central subgroup is not difficult to establish. The authors remark that U. Ray proved that if a finite group G acts freely on a product of spheres, then the only possible nonabelian composition factors are PSL_{2}(5) and PSL_{2}(9). Samuel Evens (1-AZ)

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