Minimal generating sets of transfer systems for more non-Abelian Groups

TL;DR

This paper computes the minimal generating sets and complexity of transfer systems for certain non-Abelian groups, revealing how subgroup structure influences equivariant properties.

Contribution

It extends calculations of transfer system generators and complexity to semidihedral and affine Frobenius groups, linking subgroup lattice structure to equivariant multiplicative complexity.

Findings

Computed width w(G) for semidihedral groups SD_{2^n} and affine Frobenius groups AGL(1,p^n)

Established c(D_{p^n})=⌊3n/2⌋+1 for dihedral groups of order 2p^n

Derived lower bounds for c(SD_{2^n}) based on subgroup lattice structure

Abstract

For a finite group $G$, $N_\infty$ operads encode collections of norm maps, and by work of Blumberg--Hill and Rubin their homotopy category is equivalent to the poset of $G$--transfer systems on the subgroup lattice of $G$. In \cite{ABB+25} the authors defined the \emph{width} $w(G)$ as the minimal size of a generating set for the complete $G$--transfer system and identified it with the number of conjugacy classes of proper meet irreducible subgroups of $G$, and the \emph{complexity} $c(G)$ as the maximum, over all transfer systems $T$, of the size of a minimal generating set for $T$. We compute $w(G)$ for the semidihedral groups $\SD_{2^n}$ ($n\ge 4$) and the affine Frobenius groups $\AGL(1,p^n)\cong \mathbb{F}_{p^n}\rtimes \mathbb{F}_{p^n}^\times$, extending existing calculations and highlighting how subgroup lattice structure governs equivariant multiplicative complexity. We also compute $c(D_{p^n})$ for dihedral groups of order $2p^n$ with $p$ an odd prime, establishing $c(D_{p^n})=\lfloor 3n/2\rfloor+1$, and derive the lower bound $c(\SD_{2^n})\ge\lfloor 5(n-1)/2\rfloor$.