On the uniqueness theorem
Helmut Bender · Illinois Journal of Mathematics · 1970
It is the purpose of this note to give an alternate proof of the following theorem which originally is an intermediate result of [1].THEORElYl (Felt and Thompson).Let G be a simple group of odd order all of whose proper subgroups are solvable.'Let E be an elementary abelian p-subgroup of order p8 in G. Then there is only one maximal subgroup of G which contains E.The largest part of the proof deals with the Fitting subgroup F of a maximal subgroup H of G.In 2 we consider the case that F is a p-group; necessary results about F are derived in a well known way mainly from the Transitivity Theorem (see (1.1) below) and the ZJ-Theorem (1.2).The case that F is not a p-group is treated in 3; here a very simple observation is crucial, namely that arguments in the proof of the Transitivity Theorem can be applied to certain subgroups of F.In 4, knowledge about F is used to obtain information about subgroups of H not necessarily contained in F. Finally transfer arguments finish the proof of the theorem.In the remainder of this section we introduce some notation and collect some necessary lemmas.Notation.S-subgroup Sylow p-subgroup X set of non-identity elements of X F (X) Fitting subgroup of X maximal nilpotent normal subgroup of X J (P) subgroup generated by all the abelian subgroups of maximal possible order of P r(A, ) set of A-invariant v-subgroups of Y r*(A, r) set of maximal elements of r (A, v) group of type (p, p, ..., p) elementary abelian p-group of order p r (X) _> n means that X has an elementary abelian p-subgroup of order p SCN, (P) set of abelian normal subgroups of P satisfying Cp (A) A and r(A) >_ n {...} the set (..-} the subgroup generated by In the following sections G is assumed to be a group of odd order all of whose proper subgroups are solvable.