Two new criteria for solvability of finite groups

We prove the following two new criteria for the solvability of finite groups. Theorem 1: Let G be a finite group of order n containing a subgroup A of prime power index p(s). Suppose that A contains a normal cyclic subgroup B satisfying the following condition: A/B is a cyclic group of order 2(r) for some non-negative integer r. Then G is a solvable group. Theorem 3: Let G be a finite group of order n and suppose that psi(G) >= 1/6.68 psi(C-n), where psi(G) denotes the sum of the orders of all elements of G and C-n denotes the cyclic group of order n. Then G is a solvable group. (C) 2018 Elsevier Inc. All rights reserved.