The ℱ-depth of an ℱ-projector
H. J. Schmidt · Pacific Journal of Mathematics · 1973
Let % be a saturated formation and let G be a finite solvable group with ^-projector JP.In a fundamental work, Carter and Hawkes have shown that for suitably restricted % there is a chain of j^crucial maximal subgroups of G terminating with F. It is shown here that the number of links in such a chain is an ^-invariant of G, called the gf-depth of F in G and written d%(F, G).If 4(G) is the g-length of G then, provided § is normal subgroup-closed, the inequality S % (G) ^ 2 d%(F, G) + 1 is obtained.If F is also nilpotent of nilpotency class c(F), then it is proved that 4(G) ^ d%(F, G) + c(F).If % and ξ> are two such suitable saturated formations with § = &> comparisons of the invariants d%(F, G) and d$(H, G) are made, where F and H are respectively the $-and £>-projectors of the the finite solvable group G.In particular, if H ^ F then d d (F, G) ^ d$(H, G), and if in addition d 9 (F, G) = d^H, G) then H = F.I* Introduction* In this paper all groups considered are finite and solvable.Throughout we let % be a saturated formation which is locally induced by a class of nonempty, integrated formations %(p), one for each prime p.The concepts, definitions and notation of this article are included in the above-mentioned paper [2] of Carter and Hawkes.However, we make one standard definition not found there.