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

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 $c(\SD_{2^n})\ge\lfloor 5(n-1)/2\rfloor$.