Classification of 2-generator finite metabelian p-groups having a proper \(H_ p\)-subgroup
For a group G and prime p, the subgroup of G generated by the elements of order unequal to p is called the \(H_ p\)-subgroup. Hughes conjectured that whenever \(H_ p(G)\) is a proper subgroup, it must have index p. Although in general false, the Hughes conjecture has been verified in many important classes of groups. In particular, Hogan and Kappe established its truth for finite metabelian p-groups. Additionally, they constructed an example of a 2-generator finite metabelian p-group of class p having a proper abelian \(H_ p\)-subgroup. In this paper, we generalize that construction to exhibit a family of 2-generator finite metabelian p-groups of class m, \(m\geq p\), each having a proper \(H_ p\)- subgroup of class n, \(1\leq n\leq p-2\). These values of the ordered pairs (m,n) span the full range of possibilities. We then prove that any 2- generator finite metabelian p-group having a proper \(H_ p\)-subgroup must be a homomorphic image of a group in this family. That these groups, composed ``mostly'' of elements of order p, can be constructed of arbitrary size is in contrast to the situation for finite 2-generator groups of exponent p described in a well-known result of Kostrikin. Finally, the study of these 2-generator groups enables us to obtain some interesting structural theorems for the general n-generator case.
scientific article; zbMATH DE number 4029812