An intersection property of Hall π-subgroups affecting π-length in finite π-solvable groups

摘要

0. Introduction. Suppose E is a Hall system of a solvable group, G, and U and V are subgroups of G into which I reduces [2,1.4.1,1.4.15]. Shamash [8] has shown that in this case, I also reduces into U nV. Lockett [7] has shown that if also U V = VU, then S reduces into V V as well. In their recent comprehensive volume on finite solvable groups, Doerk and Hawkes modify a proof of Lockett's result to deal with a single Hall 7i-sub-group of an arbitrary finite group G: Denote the set of Hall 7r-subgroups of G by HallIt(G), and the set of Sylow p-subgroups of G by Sylp(G). Suppose that H e Halln(G), and U and V are subgroups of G such that HnUe Hall, (I) and H n Fe HallJF). They point out [2, p. 229] that if UV=VU, then ffn Un Ve Hall %(U n V), and HnUVe Hall,, (U V). The result for H n U V is clearly of interest only when U V is a subgroup of G, i.e., when UV = FC/, but it is possible to investigate the result for H nU rV without that assumption. Doerk and Hawkes, referring to a question posed by Shamash, suggest that it is not known whether the hypothesis U V = VU can be omitted in that case, even for solvable groups [2, p. 229]. Thus let us say that a group G has Property IK if, whenever H e Hall,. (G) and U,VSG such that HnUe Halln (U) and H n V e Hall, (7), then HnU nVe Halln (U n 7). The question is, what groups have Property !?, and in particular, do all solvable groups possess that property for all choices of nl If n = {p}, a single prime, then we use lp instead of the more cumbersome I{p} to denote the Property. In [3], we have proved the following.