Finite groups containing many subgroups are solvable
The aim of this paper is to obtain sufficient conditions for the solubility of a finite group \(G\) in terms of its Hall \(\pi\)-structure. Here are three samples selected from a wealth of interesting results: Theorem 6. If \(m\) is an integer, \(2\leq m\leq\pi(G)-1\) and if \(G\) contains a Hall \(\{p_1,p_2,\dots,p_m\}\)-subgroup for all \(m\)-tuples of distinct primes in \(\pi(G)\), then \(G\) is soluble. Corollary 7. If \(G\) contains Hall \(\{p,q\}\)-subgroups for all pairs of primes dividing the order of \(G\), with the possible exception of \(\{2,3\}\) and \(\{2,5\}\) then \(G\) is soluble. Theorems 11, 12. Let \(G\) be a finite group, let \(n=\pi(G)\) and let \(k\) be an integer, \(k<\max(n,2^{n-4}+2)\). Suppose that \(G\) contains subgroups of order \(d\) for all divisors \(d\) of \(|G|\) with the exception of at most \(k\) such divisors. Then \(G\) is soluble.
scientific article; zbMATH DE number 868026