Vol. 46, No. 2, 1973

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The -depth of an -projector

H. J. Schmidt

Vol. 46 (1973), No. 2, 537–547

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 -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 Fand 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.

Mathematical Subject Classification 2000
Primary: 20D30
Received: 1 November 1971
Revised: 6 December 1971
Published: 1 June 1973
H. J. Schmidt