On the unit group of the Burnside ring for some solvable groups as a biset functor

TL;DR

This paper advances understanding of the unit group of the Burnside ring for certain solvable groups by providing explicit bases, analyzing its biset functor structure, and exploring subfunctor lattice and composition factors.

Contribution

It introduces a standard basis for the unit group for groups with abelian subgroups of index two and extends this to groups with a normal subgroup of odd index containing such abelian subgroups.

Findings

Established a basis for the unit group in specific solvable groups.

Analyzed the structure of the unit group as a biset functor.

Identified conditions for uncountably many subfunctors of the functor.

Abstract

The theory of bisets has been very useful in progress towards settling the longstanding question of determining units for the Burnside ring. In 2006 Bouc used bisets to settle the question for $p$-groups. In this paper, we provide a standard basis for the unit group of the Burnside ring for groups that contain a abelian subgroups of index two. We then extend this result to groups $G$, where $G$ has a normal subgroup, $N$, of odd index, such that $N$ contains an abelian subgroups of index $2$. Next, we study the structure of the unit group of the Burnside ring as a biset functor, $B^\times$ on this class of groups and determine its lattice of subfunctors. We then use this to determine the composition factors of $B^\times$ over this class of groups. Additionally, we give a sufficient condition for when the functor $B^\times$, defined on a class of groups closed under subquotients, has uncountably many subfunctors.