Exponential and weakly exponential subgroups of finite groups

Sabatini [L. Sabatini, Products of subgroups, subnormality, and relative orders of elements, Ars Math. Contemp., ۲۴ no. ۱ (۲۰۲۴) ۹ pp.] defined a subgroup H of G to be an exponential subgroup if x^{|G:H|} \in H for all x \in 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-trivial 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 \in G, there exists g \in G such that x^{|G:H|} \in H^g. 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(۲,q) that are wexp-solvable. We show that if \pi(n) denotes the number of primes less than n and w(n) denotes the number of primes p less than n such that PSL(۲,p) is wexp-solvable, then\[ \lim_{n \to \infty} \frac{w(n)}{\pi(n)} = \frac{۱}{۴}.\]

کلیدواژه ها:

solvable group ، simple group ، exponential subgroup