Proportion of nilpotent subgroups in finite groups and their properties

​This work introduces and investigates the function J(G) = Nil(G) / L(G) , where Nil(G) denotes the number of nilpotent subgroups and L(G) the total number of subgroups of a finite group G. The function J(G), defined on the interval (0,1], represents the proportion of nilpotent subgroups relative to the total number of subgroups, including trivial ones. It serves as a tool for analyzing structural patterns in finite groups, particularly in nonnilpotent families such as supersolvable and dihedral groups. Analytical results reveal the distribution of J(G) values across products of dihedral groups, emphasizing its density over (0,1]. Additionally, a probabilistic analysis, supported by extensive computational simulations, suggests that the sample mean of J(G) values converges in distribution to the standard normal, in line with the Central Limit Theorem, as the sample size grows. These findings contribute to the study of multiplicative functions in group theory, offering novel insights into the structural and probabilistic behavior of finite groups.