Exponential and weakly exponential subgroups of finite groups

Sabatini (2024) defined a subgroup H of G to be an exponential subgroup if x|G:H| ∈ H for all x ∈ G. Exponential subgroups are a generalization of normal (and subnormal) subgroups: all subnormal subgroups are exponential, but not conversely. Sabatini proved that all subgroups of a finite group G are exponential if and only if G is nilpotent. The purpose of this paper is to explore what the analogues of a simple group and a solvable group should be in relation to exponential subgroups. We say that an exponential subgroup H of G is exp-trivial if either $H = G$ or the exponent of G, exp(G), divides $|G:H|$, and we say that a group G is exp-simple if all exponential subgroups of G are exp-trivial. We classify finite exp-simple groups by proving G is exp-simple if and only if exp(G) = exp(G/N) for all proper normal subgroups N of G, and we illustrate how the class of exp-simple groups differs from the class of simple groups. Furthermore, in an attempt to overcome the obstacle that prevents all subgroups of a generic solvable group from being exponential, we say that a subgroup H of G is weakly exponential if, for all x ∈ G, there exists g ∈ G such that x|G:H| ∈ Hᵍ. If all subgroups of G are weakly exponential, then G is wexp-solvable. We prove that all solvable groups are wexp-solvable and almost all symmetric and alternating groups are not wexp-solvable. Finally, we completely classify the groups PSL(2,q) that are wexp-solvable. We show that if π(n) denotes the number of primes less than n and $w(n)$ denotes the number of primes p less than n such that PSL(2,p) is wexp-solvable, then limn → ∞ w(n)/π(n) = 1/4.