exponential and weakly exponential subgroups of finite groups

Sabatini [5] defined a subgroup h of g to be an exponential subgroup if x|g:h| ∈ h for all x ∈ g, in which case we write h ⩽exp 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 ⩽exp 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: (mathematical formula)