Finance

Normal subgroup

Combinatorics

Discrete mathematics

Symmetric group

Abelian group

p-group

Economics

Prime power

Organic chemistry

Order (exchange)

Prime (order theory)

Chemistry

Solvable group

Programming language

Cyclic group

Finite group

Group (periodic table)

Subgroup

Mathematics

Computer science

Integer (computer science)

Journal of Algebra

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 ps. Suppose that A contains a normal cyclic subgroup B satisfying the following condition: A/B is a cyclic group of order 2r 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 ψ(G)≥16.68ψ(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.

Two new criteria for solvability of finite groups