Vol. 29, No. 3, 1969

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On an embedding property of generalized Carter subgroups

Edward T. Cline

Vol. 29 (1969), No. 3, 491–519

If and are saturated formations, is strongly contained in (written ℰ≪ℱ) if:

  1. For any solvable group G with -subgroup E, and -subgroup F, some conjugate of E is contained in F.

This paper is concerned with the problem:

  1. Given , what saturated formations satisfy ℰ≪ℱ?

The object of this paper is to prove two theorems. The first, Theorem 5.3, shows that if 𝒯 is a nonempty formation, and = {GG∕F(G) ∈𝒯}. (F(G) is the Fitting subgroup of G), then any formation which strongly contains has essentially the same structure as in that there is a nonempty formation 𝒰 such that = {GG∕F(G) ∈𝒰}. The second, Theorem 5.8, exhibits a large class of formations which are maximal in the partial ordering . In particular, if 𝒩i denotes the formation of groups which have nilpotent length at most i, then 𝒩i is maximal in . Since for 𝒩 = 𝒩1, the 𝒩-subgroups of a solvable group G are the Carter subgroups, question (1.2) is settled for the Carter subgroups.

Mathematical Subject Classification
Primary: 20.40
Received: 12 January 1968
Published: 1 June 1969
Edward T. Cline