Characterizing Transfer Systems for Non-Abelian Groups

TL;DR

This paper studies the structure of transfer systems for non-abelian finite groups, providing explicit descriptions and expanding known lattice structures for various groups.

Contribution

It explicitly describes the properties and lattice structures of transfer systems for several non-abelian groups, including dihedral, quaternion, and dicyclic groups.

Findings

Described the width of all dihedral, quaternion, and dicyclic groups.

Established that the set of G-transfer systems forms a poset lattice.

Expanded the known transfer system lattices to include Frobenius and alternating groups.

Abstract

For a finite group $G$, the notion of a $G$-transfer system provides homotopy theorists with a combinatorial way to study equivariant objects. In this paper, we focus on the properties of transfer systems for non-abelian groups. We explicitly describe the width of all dihedral groups, quaternion groups, and dicyclic groups. For a given $G$, the set of all $G$-transfer systems forms a poset lattice under inclusion; these are a useful resource to homotopical combinatorialists for detecting patterns and checking conjectures. We expand the suite of known transfer system lattices for non-abelian groups including those which are dihedral, dicyclic, Frobenius, and alternating.