Metabelian groups of order 𝑝^{𝑛+𝑚} with commutator subgroups of order 𝑝^{𝑚}

Abstract

We consider a metabelian group G obtained by extending an abelian group H of order pn and type 1, 1, ' * by means of m operators U1, , U, of order p from a Sylow subgroup of its group of isomorphisms. Every operator of U = { U1, * * *, Urn I determines a partition of n and the fact that G is metabelian is equivalent to the requirement that no operator of U determine a partition of n in which the greatest term is greater than 2. t We require further that H be a maximal invariant abelian subgroup of G; this implies that no operator, except identity, in U determines a partition of 'P with greatest term smaller than 2. Throughout the first four sections we shall require that every operator of U, except identity, determine the partition n=2+2+1?+ * * +1. Such groups for m=3 as well as the groups such that every operator of U, except identity, determines the partition n = 2 + 1 + + 1 have been classified. t In the case m = 3 we found that there is but one group satisfying the conditions which we impose here, but the considerations necessary to show it indicated that extremely interesting results were to be found for larger values of m. In ?1 we suppose generators of G to satisfy a set of relations of a special type and are able to show that the problem of the classification of the resulting groups is exactly the problem of the classification of polynomials of degree mn in a single variable x with coefficients in the modular field, mod p, under the group of projective transformations on x with coefficients also in the modular field. This is applied in ?2 to the groups for m = 4, where some obvious properties of the groups suggest a further analysis of the relation between polynomial and group. In particular it becomes apparent that the polynomial is in most cases independent of the special form of the generating relations used in ?1; also there appears a group which belongs in the class but has no set of generators satisfying these special relations. In ?3 it is shown that the classification of the groups with central of order pn-2 is equiva-