Classifying Subgroups of Solvable Groups

Abstract

Let G and K be groups (finite or infinite). Two very general questions that might reasonably be asked are (1) does G contain an isomorphic copy of K? (2) can the isomorphic copies of K in G be classified or (in the finite case) counted? The present paper can be viewed as directed toward the second question although the first is touched on lightly. Actually it will be seen that most results concern subgroups that are isomorphic to K in a certain way. Also the answer is in terms of one dimensional cohomology. So it is perhaps accurate to say that the present paper studies relationships between the subgroup structure of a group and its low dimensional cohomology. One result which can be stated without too much terminology is the following (Corollary 3. 7). Let G = GQ^G1(J. -DG f = l be a solvable series with A,. = G,._1/Gi.. {2,. ., s] z>7 and B{ = 1, z'el, BS=A{, i&I. Suppose that if i^I and G/Gf._iZ) W and W covers B13. ., £,-_! then H(W, AJ=Q9 j=l, 2. Then (1) G has a subgroup K covering B19. ., Bs and any two such are conjugate. (2) all such<-»/7,.ej (A,./ A,. '~). This generalizes the classical theorem of P. Hall and the exact relationship to Hall's theorem is discussed in Section 3. A more typical result is Theorem 2. 4 whose conclusion reads