Overlays and group actions

Group Actions and Group Extensions

In this paper we study finite group extensions represented by special cohomology classes. As an application, we obtain some restrictions on finite groups which can act freely on a product of spheres or on a product of real projective spaces. In particular, we prove that if (Z/p)r acts freely on (S1)k , then r ≤ k.

Singularities and Group Actions

Group Actions

The symmetric groups Sn, alternating groups An, and (for n ≥ 3) dihedral groups Dn behave, by their very definition, as permutations on certain sets. The groups Sn and An both permute the set {1, 2, . . . , n} and Dn can be considered as a group of permutations of a regular n-gon, or even just of its n vertices, since rigid motions of the vertices determine where the rest of the n-gon goes. If ...

Group Actions

There are a number of results of group theory presented in Herstein [1] (the counting principles in Sections 2.5 and 2.11, Cayley’s theorem in Section 2.9 and the Sylow theorems of Section 2.12, among others) whose proofs follow similar ideas. However, it is not apparent that this is so. This handout covers the ideas necessary to tie these results together. The important idea is that of an acti...

Group Actions and Rational Ideals

We develop the theory of rational ideals for arbitrary associative algebras R without assuming the standard finiteness conditions, noetherianness or the Goldie property. The Amitsur-Martindale ring of quotients replaces the classical ring of quotients which underlies the previous definition of rational ideals but is not available in a general setting. Our main result concerns rational actions o...