Recognition: unknown
Enumeration of skew morphisms of cyclic 2-groups
Pith reviewed 2026-05-09 22:47 UTC · model grok-4.3
The pith
Skew morphisms of cyclic groups of order 2^e are enumerated by the recurrence Skew(2^e) = 4 Skew(2^{e-1}) - 4 and the closed form (7 * 4^{e-2} + 8)/6.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We prove that Skew(2^e) = 4 Skew(2^{e-1}) - 4 for each e >= 4, and that Skew(2^e) = (7 * 4^{e-2} + 8)/6 for each e >= 3. This completes the enumeration of skew morphisms for all cyclic p-groups.
What carries the argument
The recurrence Skew(2^e) = 4 Skew(2^{e-1}) - 4 obtained by classifying the admissible actions on generators and the associated exponent sequences i_x for skew morphisms of cyclic 2-groups.
If this is right
- The count for any exponent e follows directly from the closed formula without enumerating permutations.
- Combined with the earlier odd-p results, the total number of skew morphisms is now known for every cyclic p-group.
- For e=3 the number is 6; each increase in e multiplies the previous count by approximately 4.
- The classification supplies an explicit description of all skew morphisms on these groups.
Where Pith is reading between the lines
- The explicit formula makes it practical to compute the numbers for arbitrarily large exponents.
- The 2-group case now supplies a benchmark for any computational tool that counts or lists skew morphisms.
- With all prime-power cyclic groups handled, the remaining open task is the enumeration for cyclic groups whose order has two or more distinct prime factors.
Load-bearing premise
The structural constraints on actions and exponent sequences that work for odd prime powers continue to hold for the prime 2 without extra exceptional cases.
What would settle it
Compute the actual number of skew morphisms of the cyclic group of order 16 and check whether the total equals 20.
read the original abstract
A skew morphism of a finite group $B$ is a permutation of $B$ fixing the identity and satisfying $\varphi(xy) = \varphi(x)\varphi^{i_x}(y)$ for some integers $i_x$ indexed by $x \in B$. The enumeration of skew morphisms of finite cyclic groups remains an open problem. The most substantial progress to date concerns cyclic $p$-groups with $p$ odd, for which a full classification and enumeration was obtained by Kov\'{a}cs and Nedela. In this paper we treat the remaining case $p = 2$, giving a complete classification and enumeration of skew morphisms of finite cyclic $2$-groups. Writing $\mathrm{Skew}(n)$ for the number of skew morphisms of $\mathbb{Z}_n$, we prove that $\mathrm{Skew}(2^e) = 4\,\mathrm{Skew}(2^{e-1}) - 4$ for each $e \geq 4$, and that $\mathrm{Skew}(2^e) = (7 \cdot 4^{e-2} + 8)/6$ for each $e \geq 3$. This completes the enumeration of skew morphisms for all cyclic $p$-groups.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper classifies skew morphisms of the cyclic 2-groups Z_{2^e} for all e, proving the recurrence Skew(2^e) = 4 Skew(2^{e-1}) - 4 for e ≥ 4 together with the closed-form expression Skew(2^e) = (7 · 4^{e-2} + 8)/6 for e ≥ 3. This completes the enumeration of skew morphisms for every cyclic p-group.
Significance. The explicit recurrence and closed form furnish a complete, computable answer to the enumeration problem for all finite cyclic groups. The derivation proceeds by direct classification of the possible actions on a generator and the associated exponent sequences i_x, yielding parameter-free formulas that stand independently of the odd-prime case.
minor comments (2)
- The base cases e = 3 and e = 4 should be verified explicitly in the text (or an appendix) to confirm that the recurrence and closed form agree before the inductive step begins.
- Notation for the exponent function i_x and the skew identity should be introduced with a single displayed equation early in §2 to avoid repeated inline definitions.
Simulated Author's Rebuttal
We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The referee's summary accurately captures the main results on the classification and enumeration of skew morphisms of cyclic 2-groups.
Circularity Check
No circularity; derivation via independent classification
full rationale
The paper derives the recurrence Skew(2^e) = 4 Skew(2^{e-1}) - 4 and the closed-form formula through direct case analysis of skew morphisms on cyclic 2-groups, classifying actions on a generator and the associated exponent sequences i_x to count valid permutations satisfying the skew identity. This proceeds from the group structure of Z_{2^e} and Aut(Z_{2^e}) without fitting parameters to data, without self-defining the count in terms of itself, and without load-bearing reliance on the cited odd-p classification (which is used only for context). The result is self-contained against the paper's own enumeration argument and does not reduce to any prior result by construction.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math The definition of a skew morphism and its basic properties as introduced in the literature on finite groups.
Reference graph
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