Determination of all the groups of order 𝑝^{𝑚} which contain the abelian group of type (𝑚-2,1),𝑝 being any prime

Abstract

It is well known that there are just three abelian groups of order ptm which contain the group of type (n 2, 1); viz. the groups of types (n 1, 1), (n 2, 2), (m 2, 1, 1).j In what follows we shall therefore assume that the groups under consideration are non-abelian. The common abelian subgroup of type (nz 2, 1) will be represented by H. The operators of the required groups must transform those of H according to a subgroup of order p in the group of isoilnorphisms (I) of H. Our first problem is to determine all the operators of order ]) which are contained in I. Since the groups of order p4 are well known we shall always assume m > 4. The groups which contain a cyclic invariant subgroup of order p--2 are also kniown f, but we shall re-determine them in connection with the other groups since this will not materially affect the work. The group H contains one cyclic characteristic subgroup of each order of the form pa (a = 1 2, *.n 3). It also contains one non-cyclic characteristic subgroup of each order of the form pal (a,= 2, 3, .., rn2). Its other subgroups are cyclic and non-characteristic when p is odd. There are 1 1 of these of each order of the form pa2 (a2 2, 3, , -n 3) while there are p such subgroups of each of the orders p and pm-2. When p = 2 these subgroups of orders 2 and 2,n-2 are the only non-characteristic subgroups of H. All the non-characteristic subgroups of the same order are conjugate under the holomorph of H. In fact, all the operators of the highest order in the noncharacteristic subgroups of the same order are conjugate under this holomorph. We proceed to prove several theorems applying to every abelian group (A), which will be employed in what follows. The operator of the group of isomor-