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arxiv: 2511.13439 · v2 · submitted 2025-11-17 · 🧮 math.AT · math.CO

Characterizing Transfer Systems for Non-Abelian Groups

Sarah Klanderman , Chloe Lewis , Harlea Monson , Koki Shibata , Danika Van Niel This is my paper

Pith reviewed 2026-05-17 21:03 UTC · model grok-4.3

classification 🧮 math.AT math.CO
keywords transfer systemsnon-abelian groupsdihedral groupsquaternion groupsdicyclic groupsposet latticesequivariant homotopy
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The pith

The widths of transfer systems for all dihedral, quaternion, and dicyclic groups are explicitly described.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper provides a combinatorial approach to studying equivariant objects through G-transfer systems when the group G is non-abelian. It explicitly computes the width for every dihedral group, every quaternion group, and every dicyclic group. The full set of G-transfer systems is shown to form a lattice under inclusion, and new lattices are constructed for groups in the dihedral, dicyclic, Frobenius, and alternating families. These additions give researchers more concrete examples to search for patterns and to test conjectures.

Core claim

For a finite group G the notion of a G-transfer system provides homotopy theorists with a combinatorial way to study equivariant objects. We explicitly describe the width of all dihedral groups, quaternion groups, and dicyclic groups. We expand the suite of known transfer system lattices for non-abelian groups including those which are dihedral, dicyclic, Frobenius, and alternating.

What carries the argument

G-transfer system, a combinatorial structure on the subgroups of a finite group G that encodes equivariant homotopy data.

Load-bearing premise

That the standard combinatorial definition of a G-transfer system applies directly when G is non-abelian and that the enumeration of all such systems for the studied groups is complete.

What would settle it

The claim would be falsified by the discovery of a transfer system on a dihedral group whose width differs from the explicit description or by finding a system not accounted for in the lattice.

Figures

Figures reproduced from arXiv: 2511.13439 by Chloe Lewis, Danika Van Niel, Harlea Monson, Koki Shibata, Sarah Klanderman.

Figure 1
Figure 1. Figure 1: The pentagon N5. The only subgroup lattices that are non-modular are associated to non-abelian groups. An interesting property arises when one studies short edges in G-transfer systems on a non-modular lattice [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Restrictions on the non-modular lattice Sub(A4)/A4. One of the dotted edges is a short edge and the other is not. This anomaly is a result of the non-modular structure. The reader may wonder if this transfer system on Sub(A4)/A4 lifts to a transfer system on Sub(A4). The following definition and proposition determine those groups G for which transfer systems can always be lifted from Sub(G)/G to Sub(G). De… view at source ↗
Figure 3
Figure 3. Figure 3: An example of a Cpq-transfer system, a non-example of a Dp-transfer system, and a Dp-transfer system. Not every edge out of a conjugacy class will give DTC restrictions, consider the following example. Example 2.8. Consider the Frobenius group of order 20, F5. Since each conjugate copy of C4 only contains one copy of C2 and there are equal number of copies of C2 as there are copies of C4 then no edge C2 → … view at source ↗
Figure 4
Figure 4. Figure 4: A F5-transfer system with no DTC restrictions. Let us now consider DTC restrictions on Dpn . Example 2.9. Let p be an odd prime and consider the complete Dpn -transfer systems, with composition edges omitted. Note that the lattices Sub(Dpn )/Dpn build inductively, as shown in [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The lattices Sub(Dpn )/Dpn for n = 2, 3 and n arbitrary. orem 2.7 above, a transfer system on one of these lattices may have DTC restrictions; we explore these now. Restricting the edge pn C2 → (pn−1)Dp by a conjugate copy of C2 produces the edge e → pn C2, for any n. An example of this restriction in Dp3 is shown on the left in [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Two DTC restrictions in Dp3 -transfer systems. We now explore another class of dihedral groups, those of the form D2n for n ≥ 2. Example 2.10. Consider the complete D2n -transfer systems below, with composition edges omitted. Note that the lattices Sub(D2n )/D2n build inductively, as shown in [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The lattices Sub(D2n )/D2n for n = 2, 3 and n arbitrary. D8 D4 C8 D4 2C 2 2 C4 2C 2 2 4C2 C2 4C2 e [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: A DTC restriction that arises in a D8-transfer system. We now explore some DTC restrictions. The vertical edges in the wings each induce an edge from the spine to the wing. For example, (2n−1)C2 → (2n−2)C 2 2 induces the edge e → (2n−1)C2 via restriction by a conjugate copy of C2. Note the source of the first edge is the target of the second edge. In general, if (2n−i)X → (2n−i−1)Y is a vertical edge on a … view at source ↗
Figure 9
Figure 9. Figure 9: The lattices Sub(Dicpn )/ Dicpn for n = 1, 2 and n arbitrary. We now explore some DTC restrictions. Note that each copy of Dicp has p conjugate copies of C4, consider two of them, say H and K. The edge H → Dicp induces the edge C2 → K via restriction by K. This DTC restriction in a Dicp2 -transfer system is shown on the left in [PITH_FULL_IMAGE:figures/full_fig_p006_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Some DTC restrictions that arise in Dicp2 -transfer systems. Now that we have discussed some examples of transfer systems and explored the challenges that arise in the non-abelian setting, we will discuss properties of transfer systems that are relevant for studying equivariant homotopy theory [PITH_FULL_IMAGE:figures/full_fig_p006_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Compatibility condition from Theorem 2.14. In the case of B = B ∩ C, the diagram in [PITH_FULL_IMAGE:figures/full_fig_p007_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Compatibility condition when B = B ∩ C. One can saturate any transfer system T by adding in only the edges necessary for it to be saturated. This process produces the same transfer system as the following definition. Definition 2.16. Let T be a G-transfer system. The saturated hull of T, denoted Hull(T), is the smallest saturated G-transfer system that contains T. Due to the special case of the compatibil… view at source ↗
Figure 13
Figure 13. Figure 13: The subgroup lattice for the group D36. On the other hand, if one starts at Dn and divides out the same prime repeatedly until that prime is gone, this yields a path of meet-irreducible subgroups. For instance, consider the following path of edges: D36 = D2 2·3 2 ← D2 2·3 ← D2 2 . From the lattice in [PITH_FULL_IMAGE:figures/full_fig_p010_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The subgroup lattice for the group Q8. For the case when m > 1, we combine the results of Theorem 3.4 and Theorem 3.5 to get our desired result. □ Remark 3.8. The width of generalized quaternion groups was also proven independently by Evans in work to appear [Eva]. Evans’ results agree with our calculation of width and further explore the complexity of quaternion groups. 3.2. Dicyclic groups. Dicyclic gro… view at source ↗
Figure 15
Figure 15. Figure 15: The subgroup lattice for the group Dic18. Using a similar process as in Theorem 3.6 we find that the following are the meet-irreducible subgroups: Dic3 2 , Dic3 2 , C2 2·3 2 , Dic2·3, and Dic2, where the two copies of Dic3 2 are not conjugate. This example agrees with the formula of Theorem 3.9. We will now prove the theorem in full generality. Proof. By Theorem 3.2, it suffices to determine the number of… view at source ↗
Figure 16
Figure 16. Figure 16: The Hasse diagram of transfer systems for the group Cp2 . We next wish to consider a pair of transfer system lattices where the effects of the conjugacy axiom for non-abelian groups can be seen. The abelian group Cpq and the non-abelian group Dp for p, q distinct primes and p odd have very similar Sub(G)/G lattice structures. However, Dp contains conjugate subgroups. Note that there is one fewer Dp-transf… view at source ↗
Figure 17
Figure 17. Figure 17: The Hasse diagram of transfer systems for the group Cpq, for p and q distinct primes [PITH_FULL_IMAGE:figures/full_fig_p014_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: The Hasse diagram of transfer systems for the group Dp, for p an odd prime. The remaining lattice examples in this section are significantly larger and more complex than the previous ones. In [PITH_FULL_IMAGE:figures/full_fig_p015_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: The Hasse diagram of transfer systems for the group A4. △ = saturated ♡ = cosaturated ♦ = lesser simply paired (LSP) ♢ = connected ( =⇒ LSP) e 4C3 3C2 C 2 2 A4 ♢ △ · · · ♡ · · ♢ △ · · · ♡ · · ♢ △ · · · ♡ · · ♢ △ · · · ♡ · · ♢ △ · · · ♡ · · · · · · · ♢ △ · · · ♡ · · · · · · · ♢ △ · · · ♡ · · ♢ △ · · · ♡ · · ♢ △ · · · ♡ · · ♢ △ · · · ♡ · · ♢ △ · · · ♡ · · ♢ △ · · · ♡ · · ♢ △ · · · ♡ · · ♢ △ · · · ♡ · · ♢ △ … view at source ↗
Figure 20
Figure 20. Figure 20: The Hasse diagram of transfer systems for the dihedral group Dp2 , for p an odd prime. ♢ △ · ♡ · · · · · △ △ · · · · · △ △ · · · · · · · △ △ · · · · · · △ △ · · · · · · · · · · · · △ △ · · · · · · △ △ · · · · · · △ △ · · · · · · · · · · · · · · · · · · △ △ · · · · · · ♢ △ · ♡ · · · · · △ △ · · · · · · · · · · · · · · · · · · · · · · · · ♦ · ♡ · · · · · · · · · · · · · · · · ♢ · · ♡ · · · · · · · · · · · △… view at source ↗
Figure 21
Figure 21. Figure 21: The Hasse diagram of transfer systems for the group Dicp, for p an odd prime. ♢ △ · ♡ · · · · · ♢ △ · ♡ · · · · △ · ♢ · ♡ · · · · △ · ♢ · ♡ · · · · · ♢ △ · ♡ · · · · · ♢ △ · ♡ · · · · · ♢ △ · ♡ · · · · · ♢ △ · ♡ · · · · · ♢ △ · ♡ · · · · · · · · · · · ♢ △ · ♡ · · · · · ♢ △ · ♡ · · · · · · · · · · · · · · · · · · · · · · △ · ♢ · ♡ · · · · · △ ♦ · ♡ · · · · · · · · · · · · · · · · · ♢ △ · ♡ · · · · · · · · … view at source ↗
Figure 22
Figure 22. Figure 22: The Hasse diagram of transfer systems for the group F5. ♢ △ · ♡ · · · · · ♢ △ · ♡ · · · · △ · ♢ · ♡ · · · · △ · ♢ · ♡ · · · · · ♢ △ · ♡ · · · · △ · ♢ · ♡ · · · · △ · ♢ · ♡ · · · · · · · · · · △ · ♢ · ♡ · · · · · ♢ △ · ♡ · · · · · · · · · · · · · · · · · · · · · · · ♢ △ · ♡ · · · · · ♢ △ · ♡ · · · · △ · ♢ · ♡ · · · · · · · · · · · · · · · · · ♢ △ · ♡ · · · · △ · ♦ · ♡ · · · · · ♢ △ · ♡ · · · · · ♢ △ · ♡ · … view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript characterizes G-transfer systems for non-abelian finite groups. It gives explicit descriptions of the width of the poset of all such systems for every dihedral group, every quaternion group, and every dicyclic group. It further computes and displays the full lattices of transfer systems for additional non-abelian families, including Frobenius groups and alternating groups, thereby enlarging the collection of explicitly known examples.

Significance. If the enumerations are exhaustive, the explicit width formulas for the three infinite families and the new lattice diagrams supply concrete, usable data for homotopy theorists working with equivariant objects. These examples can serve as test cases for conjectures about the structure of transfer-system posets and help reveal patterns that are invisible when only abelian groups are considered.

major comments (1)
  1. [Dihedral groups] The central claim that the width is explicitly described for all dihedral groups D_n (arbitrary n) rests on a complete enumeration that simultaneously respects the conjugation action on the subgroup lattice and the transfer-system closure axioms. The manuscript should supply either a general closed-form argument or a fully documented case analysis that covers conjugacy classes for every n; any gap in this accounting would render the stated width incorrect.
minor comments (2)
  1. Define the term 'width' of a transfer-system poset at its first appearance and state whether it denotes the height, the size of the largest antichain, or another invariant.
  2. Label each lattice diagram with the specific group (or family member) it represents so that readers can match figures to the textual claims without ambiguity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment of its significance for homotopy theorists. We address the major comment below.

read point-by-point responses

  1. Referee: [Dihedral groups] The central claim that the width is explicitly described for all dihedral groups D_n (arbitrary n) rests on a complete enumeration that simultaneously respects the conjugation action on the subgroup lattice and the transfer-system closure axioms. The manuscript should supply either a general closed-form argument or a fully documented case analysis that covers conjugacy classes for every n; any gap in this accounting would render the stated width incorrect.

    Authors: The manuscript already contains a complete case analysis that covers conjugacy classes for every positive integer n. In Section 3 we separate the argument into the cases of odd and even n, which determine the conjugacy classes of reflections in D_n. For odd n the single conjugacy class of reflections is handled uniformly; for even n the two distinct classes are treated separately. In each case we enumerate the admissible assignments of subgroups to the generators while enforcing both the conjugation-equivariance condition and the transfer-system closure axioms (including the required intersections and joins). The resulting width formula is obtained directly from this exhaustive enumeration, which is verified to be exhaustive by the standard classification of subgroups of dihedral groups. We therefore maintain that the stated widths are correct for arbitrary n. revision: no

Circularity Check

0 steps flagged

No circularity: direct combinatorial description of transfer system posets

full rationale

The paper's central results consist of explicit descriptions of widths and lattices for transfer systems on families of non-abelian groups, obtained via combinatorial enumeration under the standard axioms imported from prior literature. No step reduces a claimed prediction or first-principles result to a fitted parameter drawn from the same data, a self-definition, or a load-bearing self-citation chain. The enumeration is presented as case-by-case analysis of subgroup lattices and conjugacy actions rather than a renaming or ansatz smuggled from the authors' own prior work. The derivation therefore remains self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on the established definition of G-transfer systems from equivariant homotopy theory and performs explicit enumeration for concrete groups; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption A G-transfer system is a combinatorial object attached to a finite group G that encodes equivariant homotopy data.
    Invoked throughout the abstract as the object being characterized.

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Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

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