Generalising H\"older'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 H\"older'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)}}.
Let H be a subgroup of a finite group G, and put n = [G:H]>1. If p is the least prime divisor of n, we prove that the number of subgroups K with H<= K<= G is less than c(p) n^((log_p n)/4). Here c(p) is the product of (1 - p^(-j))^(-1) over all positive integers j, multiplied by the sum of p^(-z^2) over all integers z....
We prove that Pyber's upper bound $2^{O(n\log(n))}$ for the number of minimally transitive subgroups of $S_n$ is best possible along the powers of every fixed prime, even when the groups are counted up to permutational isomorphism. As a byproduct, our construction shows that, along the powers of every fixed prime, the...
We give a computer-assisted classification, up to isomorphism, of all nontrivial finite simply reducible groups of order at most 2000. There are 7889 such groups, all of even order. We provide representatives and a table of their numbers at each order. Order 1024, which is not available as a catalogue of group objects...
We present a systematic development of the arithmetic of the class of linear orders under the ordered sum $(LO, +)$ and prove a number of new results. Our approach is based on a Euclidean algorithm for pairs of linear orders that almost additively commute. Among our results: (i.) We generalize and give unified proofs o...
We determine explicit formulae for the Betti numbers of the defining toric ideals of two families of numerical semigroups of Sally type, resolving and extending beyond several conjectures of Goel--\c{S}ahin--Singh--Srinivasan. Our approach uses Ap\'ery resolutions and their specialisations. For the family \[ \Gamma_m(n...
In 2003, H\'ethelyi and K\"ulshammer proved that if $G$ is a finite solvable group and $p$ is a prime such that $p^2$ divides $|G|$, then $k(G)\geq (49p+1)/60$, and they constructed an infinite family of groups for which this bound was claimed to be attained. We point out that the groups in this family are not Frobeniu...
T. Keller· 0 citations
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