Gassmann triples with special cycle types and applications

TL;DR

This paper explores the implications of specific cycle types in permutation groups, demonstrating their influence on subgroup conjugacy, number field determination via zeta functions, and isometric properties of Riemannian manifold coverings.

Contribution

It establishes new connections between cycle types in permutation actions and conjugacy of subgroups, with applications in number theory and differential geometry.

Findings

Almost conjugate subgroups are conjugate under certain cycle type conditions.

Number fields are uniquely determined by their Dedekind zeta functions in these cases.

Certain Riemannian manifold coverings must be isometric based on geodesic lifting properties.

Abstract

We show that if one of various cycle types occurs in the permutation action of a finite group on the cosets of a given subgroup, then every almost conjugate subgroup is conjugate. As a number theoretic application, corresponding decomposition types of primes effect that a number field is determined by the Dedekind zeta function. As a geometric application, coverings of Riemannian manifolds with certain geodesic lifting behaviors must be isometric.