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Let F be a saturated
formation and let G be a finite solvable group with F-projector F. In a fundamental
work, Carter and Hawkes have shown that for suitably restricted F there is a chain of
ℱ-crucial maximal subgroups of G terminating with F. It is shown here that the
number of links in such a chain is an F-invariant of G, called the F-depth of F in G
and written dF(F,G).
If lF(G) is the F-length of G then, provided F is normal subgroup-closed, the
inequality lF(G) ≦ 2 ⋅ dF(F,G) + 1 is obtained. If F is also nilpotent of nilpotency
class c(F), then it is proved that lF(G) ≦ dF(F,G) + o(F).
If F and H are two such suitable saturated formations with F ⊇ H, comparisons
of the invariants dF(F,G) and dH(H,G) are made, where F and H are respectively
the F− and H-projectors of the the finite solvable group G. In particular, if H ≦ F
then dF(F,G) ≦ dH(H,G), and if in addition dF(F,G) = dH(H,G) then
H = F.
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