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 \(\mathit{IE}\)-system if it contains a non-normal subgroup, so that the number \(k(G)\) of \(\mathit{IE}\)-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 \(k\) \(\mathit{IE}\)-systems is soluble, and an absolutely simple one if some simple group, but no non-soluble non-simple group, has exactly \(k\) \(\mathit{IE}\)-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 \(k\ge9\) with \(k\ne11\) is \(k(G)\) for a non-soluble non-simple group \(G\): a group \(A_5\times H\) with metabelian \(H\) and a prime from Dirichlet's theorem for large \(k\), and one of \(397\) explicit groups, checked by computer, for \(9\le k\le406\). Together with \(k(\mathrm{PSL}(2,13))=11\) this alone gives the finiteness of both sets. The exact lists also use the theorem that a non-soluble group has \(k(G)\ge7\), with equality only for \(A_5\), and \(k(G)\ne8\); 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 \(\mathrm{PSL}(2,q)\). No printed answer to the problem was found in the sources we searched. This is an unrefereed note.