System permutability in finite solvable groups

摘要

The central concept considered is the following.;Definition. A subgroup H of a finite solvable group G is said to be system permutable in G if there is a Hall system $Sigma$ of G such that HS is a subgroup of G for all $SinSigma.$.;All groups considered are finite and solvable. One focus is the study of system permutable subgroups in Lagrangian groups. A group is Lagrangian if it possesses subgroups of all possible orders. Various subclasses of the class of all Lagrangian groups have been extensively studied.;One well studied class is ${cal Y}={G vert$ for all $Hle G, dVert G:Hvert,$ there exists $Kle G$ such that $Hle K$ and $vert G:Kvert=d}$. A result of A. Mann says H is system permutable if and only if $H=bigcapsb{i}Hsb{i}$, where $vert G:Hsb{i}vert=psbsp{i}{asb{i}}$ is a prime power and $psb{i}not= psb{j}$ if $inot= j.$ McLain established that $Gin{cal Y}$ if and only if every subgroup of G satisfies this condition. We see that $Gin{cal Y}$ if and only if every subgroup of G is system permutable. With this motivation, we address the problem of characterizing all of the usual Lagrangian classes by conditions involving system permutability.;We introduce the following.;Definition (Brewster). A subgroup H of G is locally system permutable in G if all the Sylow subgroups of H are system permutable in G.;It is not difficult to see that system permutability implies local system permutability. We address the.;Question. Does local system permutability imply system permutability?;Generally, the answer is "no." We present an example, originally discussed by J. Alperin in connection with a related notion. We give hypotheses on the structure of G or embedding of H in G sufficient to guarantee H is system permutable provided it is locally system permutable.;Another property that appears to be weaker than system permutability is introduced. Suppose $Hle G$ and for each prime p, there exists $Gsb{p}in{rm Syl}sb{p}(G)$ such that $HGsb{p}$ is a subgroup of G. Whether this is weaker than system permutability remains an open problem, but we prove some results about this situation.