On the lattice of J-subnormal subgroups

Fitting classes and lattice formations II

“…Our main aim in this paper is to prove that the following result, for an ^"-Fitting class 3C, holds: If W is an "-injector of a group G and H is an ^"-subnormal subgroup of G, then H D W is an "-maximal subgroup of H. In fact, this property characterizes ^"-Fitting classes (see Theorem 3.9 and Proposition 3.3), as the existence of injectors characterizes Fitting classes. The result obtained in [4,Theorem 4.5] appears now as one particular case.…”

Nilpotent-like fitting formations of finite soluble groups

“…(1) equivalent to (2). First observe that if T is an s n -closed saturated formation satisfying (2), then it is a Fitting class since every subnormal subgroup is ^-subnormal.…”

“…Arguing as in (2) and (3) of [2,Theorem 3.3], we obtain that if p and q are two prime numbers such that p € charF(g), then charF(p) = charF(g). Now, by [5, IV,Lemma 3.16], F(p) is a subgroup-closed Fitting class for every prime p. Hence, if p,g G P and p€ charF(g), then 5 P C F{q).…”

“…We 428 A. Ballester-Bolinches, M.D. Perez-Ramos and A. Martinez-Pastor [2] denote by IP the set of all prime numbers and, for a prime p, C p is the cyclic group of order p. For a group G, ir(G) denotes the set of all prime divisors of \G\. If X is a class of groups, the characteristic of X is charA" = {p € P | C p e X}.…”

“…The class of groups whose members are a direct product of Hall subgroups corresponding to a given partition of the set of all primes is a subgroup-closed saturated formation enjoying many interesting properties of nilpotent type. In fact, this family of formations was characterised in [2] as the subgroup-closed saturated formations T for which the set of all ^"-subnormal subgroups of each group G is a sublattice of the subgroup lattice of G. Hence the members of this family are called lattice formations.…”