The Laitinen Conjecture for finite non-solvable groups

Abstract

AbstractFor any finite groupG, we impose an algebraic condition, theGnil-coset condition, and prove that any finite Oliver groupGsatisfying theGnil-coset condition has a smooth action on some sphere with isolated fixed points at which the tangentG-modules are not isomorphic to each other. Moreover, we prove that, for any finite non-solvable groupGnot isomorphic to Aut(A6) or PΣL(2, 27), theGnil-coset condition holds if and only ifrG≥ 2, whererGis the number of real conjugacy classes of elements ofGnot of prime power order. As a conclusion, the Laitinen Conjecture holds for any finite non-solvable group not isomorphic to Aut(A6).