Smith equivalence and finite Oliver groups with Laitinen number 0 or 1

Smith equivalence and finite Oliver groups with Laitinen number 0 or 1 Krzysztof

In 1960, Paul A. Smith asked the following question. If a finite group G acts smoothly on a sphere with exactly two fixed points, is it true that the tangent G-modules at the two points are always isomorphic? We focus on the case G is an Oliver group and we present a classification of finite Oliver groups G with Laitinen number aG = 0 or 1. Then we show that the Smith Isomorphism Question has a...

Smith Equivalence of Representations for Finite Perfect Groups

Using smooth one-fixed-point actions on spheres and a result due to Bob Oliver on the tangent representations at fixed points for smooth group actions on disks, we obtain a similar result for perfect group actions on spheres. For a finite group G, we compute a certain subgroup IO′(G) of the representation ring RO(G). This allows us to prove that a finite perfect group G has a smooth 2–proper ac...

Categorical Equivalence of Finite Groups

In [1] Bergman and Berman define the notion of categorically equivalent algebras. In [6] we introduce minimal relational sets and point out the relationship between categorical equivalence of algebras and minimal relational sets corresponding to the algebras. In this paper we shall describe the minimal relational sets assigned to finite groups, see Theorem 3.1. Moreover, by using this descripti...

Finite Groups With a Certain Number of Elements Pairwise Generating a Non-Nilpotent Subgroup

Equivalence classes of functions between finite groups

Two types of equivalence relation are used to classify functions between finite groups into classes which preserve combinatorial and algebraic properties important for a wide range of applications. However, it is very difficult to tell when functions equivalent under the coarser (“graph”) equivalence are inequivalent under the finer (“bundle”) equivalence. Here we relate graphs to transversals ...