"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 2. Let G be a finite group of order n and suppose that ψ(G) ≥ 1/6.68 ψ(Cn), where ψ(G) denotes the sum of the orders of all elements of G and Cn denotes the cyclic group of order n. Then G is a solvable group.