Note on Caranti's Method of Construction of Miller groups

TL;DR

This paper examines Caranti's module theoretic methods for constructing Miller p-groups, showing their limitations and providing conditions under which they are effective.

Contribution

It identifies the limitations of Caranti's methods and offers a sufficient condition for their successful application to special Miller p-groups.

Findings

Caranti's methods do not always work for constructing Miller p-groups.

A specific condition ensures the effectiveness of Caranti's methods.

Examples demonstrate the limitations and applicability of the methods.

Abstract

The non-abelian groups with abelian group of automorphisms are widely studied. Following Earnley, such groups are called Miller groups, since the first example of such a group was given by Miller in 1913. Many other examples of Miller $p$-groups have been constructed by several authors. Recently, A. Caranti [{\it Israel J. Mathematics {\bf 205} (2015), 235-246}] provided module theoretic methods for constructing non-special Miller $p$-groups from special Miller $p$-groups. By constructing examples, we show that these methods do not always work. We also provide a sufficient condition on special Miller $p$-group for which the methods of Caranti work.