Minimal generating sets of transfer systems for more non-Abelian Groups
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arXiv
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| Format: | Preprint |
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2026
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| author | Chakravarthy, Bheemarasetty Ghosh, Surojit |
| author_facet | Chakravarthy, Bheemarasetty Ghosh, Surojit |
| contents | 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 $c(\SD_{2^n})\ge\lfloor 5(n-1)/2\rfloor$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_04780 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Minimal generating sets of transfer systems for more non-Abelian Groups Chakravarthy, Bheemarasetty Ghosh, Surojit Combinatorics Algebraic Topology 06B15 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 $c(\SD_{2^n})\ge\lfloor 5(n-1)/2\rfloor$. |
| title | Minimal generating sets of transfer systems for more non-Abelian Groups |
| topic | Combinatorics Algebraic Topology 06B15 |
| url | https://arxiv.org/abs/2605.04780 |