OnF-Subnormal Subgroups andF-Residuals of Finite Soluble Groups

Fitting classes and lattice formations I

“…By [8,Remark] we know that G 6 ^i if and only if G has a normal Hall a (p/-subgroup, for every prime number pen, and a normal Hall n -subgroup. Now the result is easily deduced.…”

“…With some restrictions on the sets of primes a(p) which define #', it is possible to obtain a stronger form of above-mentioned property. The formations which appear were also studied in [8] with full characteristic. Assume also that the following property holds: if q e cr(p), then a(q) C a{p), for every pair of prime numbers p,q e n. Then G e& if and only ifGe ^ and G has a normal Hall a (p)'-subgroup for every prime number p.…”

A family of dominant Fitting classes of finite soluble groups

“…{3,5}. Let V 5 be an irreducible and faithful E 3 = fC 3 ]C 2 -module over GF (5) and V 3 an irreducible and faithful [V 5 ]E 3 -module over GF (3). Construct G = [V 3 ]([V 5 ]E 3 ).…”

“…In [4], the formations & whose minimal non-^"-groups are Schmidt groups (that is, minimal non-nilpotent groups) are described. In [3], we met formations characterized through the existence of normal 7r(p)-complements in the groups of the class, for every prime p, where n(p) is a set of primes containing p.…”

“…The common property of these formations & considered in the above references [3] In the following, all groups considered are finite and soluble. Moreover we say that a Fitting class J 5 " is dominant if & is a Fitting class which is dominant in 5?.…”