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On Camina Groups of Prime Power Order

1996; Elsevier BV; Volume: 181; Issue: 3 Linguagem: Inglês

10.1006/jabr.1996.0146

1090-266X

Rex Dark, Carlo Maria Scoppola,

graph theory and CDMA systems

This paper is devoted to the class of finite p-groups with the property that every nontrivial coset of the commutator subgroup consists of conjuw x gate elements. This class, introduced by Camina C , has been studied by w x w x w x w x Macdonald, Mann, and Scoppola Mc1 , Mc2 , M , MS , and also, in the w x w x case of class 2, by Beisiegel and Verardi Be , V under the name of semiextraspecial p-groups. Our main result here shows that the nilpotency class of such a group is bounded by three. On the way, we will also obtain a number of structural properties of these groups. Results from the papers of Mcdonald, Mann, and Scoppola mentioned above will be stated without proof. w x It should be noted that the sketch of proof given for Lemma 2.1 in Mc2 does not appear to be conclusive. Not much of the literature seems to be w x affected by this possible gap, and those few results, in MS , that are affected, are in fact superseded by our results here. The best result that we w x can prove in the direction of Lemma 2.1 in Mc2 is Lemma 5 below.