The number of groups of cubefree order

Abstract: Generalising Hölder's classical group enumeration for squarefree orders (1895), we provide an exact formula for the number of isomorphism types of groups of a given cubefree order. After more than 130 years, this is the first such formula that covers significantly more orders than the squarefree ones (83% versus 61% of all integers). Like Hölder's formula, ours is combinatorial: it can be evaluated from the prime factorisation of the order by arithmetic operations and table look-ups, without constructing a single group. The structure of our formula leads to counting formulas for natural subclasses of cubefree groups, with applications in computational group theory. We also derive new asymptotic results. Blackburn et al. (2007) conjectured that the number gnu(n) of groups of cubefree order n satisfies gnu(n)<n^2. We show that gnu(n)\leq n^{2+o(1)}, which improves the bound gnu(n)<n^8 recorded in their survey, and we prove that the exponent 2 is best possible, that is, gnu(n)\geq n^{2-o(1)} for infinitely many cubefree n. Lastly, we show that a much stronger form of the conjecture holds for almost every cubefree order, namely, \gnu(n)\leq (\log n)^{(\log\log n)^{O(1)}}.

• The paper establishes the first exact, combinatorially evaluable formula for the number of groups of every cubefree order, extending Hölder’s squarefree-order result and covering approximately 83% of all integers.
• The authors prove the sharp asymptotic scale \(\mathrm{gnu}(n)=n^{2+o(1)}\) for cubefree orders, show the exponent 2 is attained infinitely often, and demonstrate that almost all such integers have only subpolynomially many groups.
• The formulas are implemented and verified in GAP, enabling rapid enumeration and structural classification while supporting applications to computational group databases, construction-by-ID, and counts with prescribed invariants.

Overview

This paper by Dietrich and Jefferies resolves a long-standing enumeration problem in finite group theory: it provides the first exact formula for $\mathrm{gnu}(n)$, the number of isomorphism types of groups of cubefree order $n$. This generalises Hölder's classical 1895 formula for squarefree orders, and covers approximately $83\%$ of all integers, compared to roughly $61\%$ for the squarefree case. Like Hölder's formula, the new formula is combinatorial in a precise sense: it can be evaluated from the prime factorisation of $n$ using integer arithmetic, divisibility tests, and look-ups in fixed tables, without constructing any group. Beyond the exact formula, the paper establishes asymptotic results that settle the exponent in a conjecture of Blackburn–Neumann–Venkataraman up to subpolynomial factors, and proves that this exponent is best possible.

Structural reduction

The formula rests on known structure theory for cubefree groups. Every group $G$ of cubefree order decomposes as $G = A \times L$, where $A$ is trivial or isomorphic to $\mathrm{PSL}_2(p)$ for a prime $p > 3$, and $n$0 is solvable. For solvable $n$1, the Frattini subgroup $n$2 is nilpotent and squarefree (hence cyclic), its order divides $n$3 where $n$4, and the isomorphism type of $n$5 is uniquely determined by $n$6 together with $n$7. The Frattini quotient is Frattini-free, hence of the form $n$8 with $n$9 the socle and $83\%$0; Gaschütz's theorem reduces isomorphism testing to conjugacy of complements.

Writing $83\%$1 for the number of solvable Frattini-free groups of order $83\%$2 with socle order $83\%$3, the authors reduce the general count via

$83\%$4

where $83\%$5 is the product of primes whose squares divide $83\%$6 (candidates for the Frattini order) and $83\%$7 records possible simple direct factors. The heart of the paper is then an explicit evaluation of $83\%$8: since two complements yield isomorphic groups exactly when they are conjugate in $83\%$9, the problem becomes one of counting conjugacy classes of suitable subgroups of $61\%$0.

The odd-order formula

For odd $61\%$1, all groups are solvable by the Feit–Thompson theorem, and every socle complement $61\%$2 is abelian. The main formula (Theorem D) sums over:

• Frattini order $61\%$3 and socle order $61\%$4: the outer sums index structural invariants of the groups being counted.
• Complement type $61\%$5: an abelian group of order $61\%$6, specified by abelian invariants.
• Projection tuples $61\%$7: each projection $61\%$8 is replaced by a canonical conjugacy-class representative from a set $61\%$9 built from scalar subgroups $n$0, diagonal subgroups of $n$1, and Singer cycle subgroups $n$2. Each representative carries an involutory normaliser action, generating an elementary abelian $n$3-group $n$4.
• Cauchy–Frobenius averaging: the summand counts $n$5-orbits on the set $n$6 of subdirect products, reduced via primary decomposition to closed formulas for the fixed-point cardinalities $n$7, which depend only on small integers $n$8, $n$9, $G$0 recording ranks of Sylow components and eigenspaces.

The worked example $G$1 illustrates the mechanics and recovers $G$2. Restricted to squarefree $G$3, the formula collapses to Hölder's expression — though evaluated as a sum rather than in closed form. The authors argue, citing the twenty-summand formula for groups of order $G$4, that a uniform closed form analogous to Hölder's is unlikely to exist for general cubefree orders.

The even-order case

Even orders are substantially harder because the complement $G$5 need not be abelian. Writing $G$6 with $G$7 odd, the authors show $G$8 where $G$9 is a characteristic abelian Hall $G = A \times L$0-subgroup and $G = A \times L$1 is a Sylow $G = A \times L$2-subgroup ($G = A \times L$3). Counting reduces to orbits on pairs $G = A \times L$4 under simultaneous conjugation. The case $G = A \times L$5 extends the odd machinery verbatim; $G = A \times L$6 reduces to counting involutions via a table of values $G = A \times L$7 depending only on six "column types" of the projections; $G = A \times L$8 requires homomorphisms $G = A \times L$9 modulo $A$0, handled by Möbius inversion over the subgroup lattice of $A$1 combined with table look-ups. An appendix supplies the technical combinatorial evaluations confirming that even this case satisfies the combinatoriality definition.

Asymptotic results

Three theorems concern the growth of $A$2:

• Upper bound: $A$3 for all cubefree $A$4. This improves the previously best bound $A$5 and asymptotically improves the $A$6 bound of Kumar–Venkataraman for solvable groups. The proof bounds the number of triples $A$7 by $A$8, controls the choices of $A$9 via a uniform bound of $\mathrm{PSL}_2(p)$0 on relevant conjugacy classes per column, and proves the key estimate $\mathrm{PSL}_2(p)$1 by factoring over columns.
• Sharpness: there is an infinite set of cubefree integers with $\mathrm{PSL}_2(p)$2, so the exponent $\mathrm{PSL}_2(p)$3 cannot be lowered. The construction squares a squarefree $\mathrm{PSL}_2(p)$4 with many nontrivial Hölder factors and exhibits enough groups with socle $\mathrm{PSL}_2(p)$5 factors. Concretely, the authors compute $\mathrm{PSL}_2(p)$6 for a specific $\mathrm{PSL}_2(p)$7 near $\mathrm{PSL}_2(p)$8, refuting the pattern suggested by exhaustive verification up to $\mathrm{PSL}_2(p)$9.
• Typical behaviour: for almost all cubefree $p > 3$0, $p > 3$1, hence $p > 3$2. This uses the decomposition theory of Dietrich–Wilson for "most" orders, bounding the count by orbit sizes of homomorphisms into automorphism groups of cyclic Hall subgroups.

Together these results confirm the conjectured exponent $p > 3$3 in Blackburn–Neumann–Venkataraman's conjecture up to $p > 3$4 factors, while showing the conjecture's stronger pointwise form fails dramatically on sparse subsequences yet holds in a much stronger averaged sense.

Implementation and verification

The formulas are implemented in GAP and will appear in the Cubefree package. Performance figures are striking: enumerating the $p > 3$5 groups of order $p > 3$6 takes $p > 3$7 seconds, and the $p > 3$8 groups of a thirteen-prime-squared order near $p > 3$9 take milliseconds. Cross-checks against the SmallGroups Library (all cubefree orders up to $n$00), Hölder's formula (squarefree $n$01), and the existing NumberCFGroups function (all cubefree $n$02, plus selected larger orders) all agree. The implementation incorporates redundancy-reduction optimisations beyond the raw formulas, and the authors note assistance from Claude Opus 5 in implementing them; the mathematical proofs themselves do not rely on computation.

Applications

Each outer summation is indexed by a structural invariant (simple factor, Frattini order, socle order), so truncating sums yields counts of groups with prescribed invariants. This is precisely the clustering data required for construction-by-ID and identification schemes of the kind used in extensions of the SmallGroups library. The authors also derive a counting formula for supersolvable cubefree groups by restricting the admissible projections, and state that identification and construction-by-ID for cubefree groups based on these formulas is left to future work.

Limitations and open questions

Several caveats are stated explicitly. The formula is combinatorial but not closed: no analogue of Hölder's compact product formula is known, and the authors doubt one exists given the congruence-laden case distinctions inherent to the problem. Evaluation cost grows with the number of prime divisors of $n$03, particularly those with $n$04; the Erdős–Pálfy factorisation trick applies only to odd orders, as the associated graph is always connected for even $n$05. The asymptotic upper bound leaves a polynomial-factor gap against the conjectured $n$06, and the almost-all result does not quantify the density of exceptional orders. Whether the cluster structure induced by the formula's summation indices suffices for efficient identification algorithms remains unaddressed.

Conclusion

The paper delivers the first exact, combinatorially evaluable formula for the number of groups of arbitrary cubefree order, genuinely extending Hölder's nineteenth-century result, together with a near-sharp asymptotic analysis showing $n$07 with the exponent $n$08 attained infinitely often, and polylogarithmic typical behaviour. The combination of exact enumeration, verified implementation, and structural indexing makes the work directly applicable to computational group theory, while leaving open the existence of closed forms and the precise polynomial refinement of the upper bound.