### Polynomial-time normalizers

*Eugene M Luks, Takunari Miyazaki*

#### Abstract

For an integer constant d > 0, let
Γ

_{d}denote the class of finite groups all of whose nonabelian composition factors lie in S_{d}; in particular, Γ_{d}includes all solvable groups. Motivated by applications to graph-isomorphism testing, there has been extensive study of the complexity of computation for permutation groups in this class. In particular, the problems of finding set stabilizers, intersections and centralizers have all been shown to be polynomial-time computable. A notable open issue for the class Γ_{d}has been the question of whether normalizers can be found in polynomial time. We resolve this question in the affirmative. We prove that, given permutation groups G, H ≤ Sym(Ω) such that G ∈Γ_{d}, the normalizer of H in G can be found in polynomial time. Among other new procedures, our method includes a key subroutine to solve the problem of finding stabilizers of subspaces in linear representations of permutation groups in Γ_{d}.Full Text: PDF PostScript