On Trofimov's Group-Theoretic Numbers (Kourovka Notebook Problem 2.78): The Soluble and the Absolutely Simple Numbers Form Finite Sets

P. I. Trofimov (Kourovka Notebook, Problem 2.78, 1967) calls the set of all subgroups of one order of a finite group G an IE-system if it contains a non-normal subgroup, so that the number k(G) of IE-systems is the number of orders of non-normal subgroups of G. A positive integer k is a soluble group-theoretic number if every finite group with exactly k IE-systems is soluble, and an absolutely simple one if some simple group, but no non-soluble non-simple group, has exactly k IE-systems; Trofimov asks whether these two sets are finite, and whether some number is composite (attained by no simple group) but not soluble. We give a partial answer: both sets are finite (the soluble numbers are exactly 1, 2, 3, 4, 5, 6, 8, and the absolutely simple numbers are 7 and possibly 11); the second question stays open (a composite non-soluble number would exceed 323). Our main result is that every k ≥ 9 with k ≠ 11 is k(G) for a non-soluble non-simple group G: a group A5 × H with metabelian H and a prime from Dirichlet's theorem for large k, and one of 397 explicit groups, checked by computer, for 9 ≤ k ≤ 406. Together with k(PSL(2,13)) = 11 this alone gives the finiteness of both sets. The exact lists also use the theorem that a non-soluble group has k(G) ≥ 7, with equality only for A5, and k(G) ≠ 8; it is in print (Zhong 2005; Shi, Zhang and Liang 2015), we know these papers only from their reviews and abstracts, and we include a proof based on Thompson's classification of the minimal simple groups. The bound 323 comes from computational data on the groups PSL(2,q). No printed answer to the problem was found in the sources we searched. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: KOU-2.78 (Kourovka Notebook Problem 2.78, P. I. Trofimov, 1967).