On an embedding property of generalized Carter subgroups

Edward Cline
1969 Pacific Journal of Mathematics  
If if and J^"~are saturated formations, ί §Γ is strongly contained in &~ (written gf « jr) if : (1.1) For any solvable group G with if -subgroup E, andj^~subgroup F, some conjugate of E is contained in F. This paper is concerned with the problem: (1.2) Given g% what saturated formations J^~satisfy g 7 « J^"? The object of this paper is to prove two theorems. The first, Theorem 5.3, shows that if J7" is a nonempty formation, and ξ?={G\GIF(G)ejT} 9 (F(G) is the Fitting subgroup of G), then any
more » ... mation ^~ which strongly contains 8Γ has essentially the same structure as c & in that there is a nonempty formation ^ such that &~ = {G \ G/F(G) e %f}. The second, Theorem 5.8, exhibits a large class of formations which are maximal in the partial ordering «. In particular, if ^A ri denotes the formation of groups which have nilpotent length at most i, then ^V* is maximal in <. Since for <yV" -^yf^1, the ^/^-subgroups of a solvable group G are the Carter subgroups, question (1.2) is settled for the Carter subgroups. EDITORS H. ROYDEN
doi:10.2140/pjm.1969.29.491 fatcat:fij24smkbvdldfkr4uos2qbzty