A Meier-Wunderli theorem for \(H_ p\)-subgroups
For a group G and a 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. Much of this work is based on results obtained for the Burnside problem. The work of Hughes and Thompson examining the relationship between \(H_ p\)-subgroups and Frobenius kernels has extended the investigations into questions on the structure of \(H_ p\)-subgroups. In this paper we prove an analogue to a well-known result of Meier-Wunderli on finite metabelian groups of exponent p. Our result establishes sharp bounds on the nilpotency class of proper \(H_ p\)-subgroups in finite metabelian p-groups. In particular, we show that the class of \(H_ p(G) \leq p\) in the general case and that the class of \(H_ p(G) \leq p-2\) when G is a 2-generator group. A significant consequence of the theorem is a proof that the \(H_ p\)-subgroups in this family of groups must be both (p-1)-Engel and regular p-groups.
scientific article; zbMATH DE number 3865561
scientific article