This paper proves: Let F be a saturated formation containing U . Suppose that G is a group with a normal subgroup H such that G/H ∈ F . (1) If all maximal subgroups of any Sylow subgroup of F ∗(H) are c-supplemented in G, then G ∈ F ; (2) If all minimal subgroups and all cyclic subgroups with order 4 of F ∗(H) are c-supplemented in G, then G ∈ F .

We prove that the Harish-Chandra’s Schwartz space associated with a discrete subgroup of a semisimple Lie group is a dense subalgebra of the reduced C∗-algebra of the discrete subgroup. Then, we prove that the reduced C∗-norm is controlled by the norm of the Harish-Chandra’s Schwartz space. This inequality is weaker than property RD and holds for any discrete group acting isometrically, properl...

‎In 1970‎, ‎Menegazzo [Gruppi nei quali ogni sottogruppo e intersezione di sottogruppi massimali‎, ‎ Atti Accad‎. ‎Naz‎. ‎Lincei Rend‎. ‎Cl‎. ‎Sci‎. ‎Fis‎. ‎Mat‎. ‎Natur. 48 (1970)‎, ‎559--562.] gave a complete description of the structure of soluble $IM$-groups‎, ‎i.e.‎, ‎groups in which every subgroup can be obtained as intersection of maximal subgroups‎. ‎A group $G$ is said to have the $FM$...

in this paper we enumerate fuzzy subgroups, up to a natural equivalence, of some finite abelian p-groups of rank two where p is any prime number. after obtaining the number of maximal chains of subgroups, we count fuzzy subgroups using inductive arguments. the number of such fuzzy subgroups forms a polynomial in p with pleasing combinatorial coefficients. by exploiting the order, we label the s...

Increased abundance of Gardnerella vaginalis and sialidase activity in vaginal fluid is associated with bacterial vaginosis (BV), a common but poorly understood clinical entity associated with poor reproductive health outcomes. Since most women are colonized with G. vaginalis, its status as a normal member of the vaginal microbiota or pathogen causing BV remains controversial, and numerous clas...

I present a simple numerical model based on iteratively updating subgroups of a population, individually modeled by nonnegative real numbers, by a constant decay factor; however, at each iteration, one group is selected to instead be updated by a constant growth factor. I discover a relationship between these variables and their respective probabilities for a given subgroup, summarized as the v...