An arXiv preprint gives an explicit asymptotic ceiling for the number of minimally transitive subgroups of S_n, the group of all permutations of n points. It also shows that, when n runs through powers of any fixed prime, the count keeps the same n log(n) scale even after actions related by permutational isomorphism are counted as one.
The upper bound has the form 2 raised to the power [(3 + log(3)/3 + o(1)) n log(n)]. The notation describes the rate of growth as degrees become large; the paper does not specify the o(1) term or exact behavior at finite n.
A ceiling with an unresolved constant
That is a ceiling on the number of subgroups, not an exact total. The unresolved part is the leading constant in the exponent: the paper places the two open asymptotic constants between 1/2 and 3 + log(3)/3, or approximately 3.52832 at the upper end.
To establish the ceiling, the proof counts maximal solvable subgroups, bounds their orders, and controls the generators of subgroups that are minimal with respect to their orbits. It then applies Stirling's formula before arriving at the final asymptotic estimate.
Prime powers supply the matching scale
The lower-bound construction begins at prime-power degrees and uses an imprimitive transitive action on V × F_p. In practical terms, the action is organized around a block system, which provides the structure for generating many different group actions.
At the algebraic core, the construction encodes K(b) as the ideal generated by the coordinates of Ψ(b). The paper uses this encoding in its construction of many distinct permutational isomorphism classes.
For each fixed prime p, and for degrees n that are powers of p, the number of permutational isomorphism classes is at least 2 raised to the power [(1/p - o(1)) n log(n)]. That lower bound has the same n log(n) exponent as the general upper-bound scale, although the result does not identify a common exact leading constant.
The largest examples grow exponentially
The same line of construction also produces a minimally transitive group of order n p^(n/p - p) for every p-power degree satisfying n >= p^2. The formula is a constructive lower bound on group size, not a claim that the exact maximum has been found.
Along powers of each fixed prime, the asymptotic picture is that the maximum order grows as 2^(Theta(n)), exponential in the degree n.
The graph count has a boundary
The group count feeds into a related question about symmetry in graphs. The paper finds that the numbers of labelled vertex-transitive graphs and labelled vertex-transitive digraphs of order n are both 2^(Theta(n log(n))).
This corollary is obtained by combining minimally transitive automorphism subgroups with orbital counting, a way of organizing the graph structures associated with group actions.
But the conclusion is deliberately narrower than a claim about nearly all symmetric graphs. These estimates concern labelled objects and do not show that asymptotically almost all labelled vertex-transitive graphs or digraphs are Cayley; unlabelled counts are not settled by the result.
The unweighted number of minimally transitive groups is also too large for the proposed route to an unlabelled digraph result. A weighted estimate remains open, leaving that stronger conclusion unresolved.
The work is listed as arXiv:2608.25792v1 and dated 26 Aug 2026.
Paper data and sources
Original title: Asymptotic enumeration of minimally transitive permutation groups
Authors: Binzhou Xia, Shasha Zheng
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text