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Note on Caranti's Method of Construction of Miller groups

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.