Two new criteria for solvability of finite groups (Q1663520)

The authors prove two results, that can be called `criteria for solvability'. The first result says that if a finite group \(G\) contains a subgroup \(A\) of prime power index containing a normal cyclic subgroup \(B\) such that \(A/B\) is a cyclic group of order \(2^r\), then \(G\) is solvable. The particular case when \(r\leq 1\) is a corollary of a theorem proved by \textit{B. Huppert} and \textit{N. Itô} [Math. Z. 61, 94--99 (1954; Zbl 0056.02202)]. The second result concerns the behaviour of the sum \(\psi(G)\) of the orders of all elements of \(G.\) If \(|G|=n\) and \(\psi(G)\leq \psi(C_n)/6.68,\) then \(G\) is solvable. The authors think that their result is not best possible; indeed they conjecture that if \(G\) is a non-solvable group of order \(n,\) then \[\phi(G)/\psi(C_n)\leq \psi(A_5)/\psi(C_{60})=211/1617.\] This conjecture has been recently proved by \textit{M. Baniasad Azad} and \textit{B. Khosravi} [J. Algebra 516, 115--124 (2018; Zbl 1446.20037)].