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Let G be a finite group and C(G) denote the set of all non-normal non-cyclic subgroups of G. In this paper, the function δc(G) = 1 |G| P H∈C(G) |H| is introduced. In fact, we prove that, if δc(G) ≤ 10/3, then either G ∼=A5, or G is solvable. We also find some examples of finite groups G with δc(G) ≤ 10/3. [ABSTRACT FROM AUTHOR]

In this article, we prove that if R is a proper alternative ring whose additive group has no 3-torsion and whose non-zero commutators are not zero-divisors, then R has no zero-divisors. It follows from a theorem of Bruck and Kleinfeld that if, in addition, the characteristic of R is not 2, then the central quotient of R is an octonion division algebra over some field. We include other characterizations of octonion division algebras and we also deal with the case where (R , +) has 3-torsion. [ABSTRACT FROM AUTHOR]