On Camina Groups of Prime Power Order

Generalizing Camina groups and their character tables

“…In [9] and [10], Macdonald and Mann proved that if G is a Camina 2-group, then its nilpotence class is 2. In the important paper [1], Dark and Scoppola proved that if G is a Camina p-group with p odd, then the nilpotence class of G is at most 3. It follows that if G is a nilpotent, generalized Camina group, then G is isoclinic to a nilpotent Camina group H. It is known that H must be a p-group for some prime p. It is not di‰cult to see that any Camina group isoclinic to G will be a p-group for the same prime p, and we will say that p is the prime associated with G. (It is not di‰cult to show that p is well defined.) Using the isoclinism, we obtain bounds on the nilpotence class of nilpotent, generalized Camina groups.…”

Groups with two extreme character degrees and their normal subgroups

“…These bounds for the nilpotency class are sharp, as is shown by the p-groups of maximal class and order p 4 and p 5 , respectively. Since the class of the p-groups with either of the conditions on normal subgroups is very small, it makes sense to study them in separate cases by fixing the class.…”

“…Also, we have developed our study of these groups a little further than necessary for our initial purpose and have obtained the following important theorem, which shows that in many of the cases in Theorem F it is not only the index of the centre of G that is small, but even the order of G. Since the p-groups of order at most p 6 are completely determined up to isomorphism (see [6,15]), it would be routine to classify the groups in either (i) or (ii) above. On the other hand, we will give examples showing that the order of G cannot be bounded if G has the weak condition on normal subgroups, class 3 and |G : Z(G)| = p 3 or p 4 . We want to emphasize that the proofs of Theorems D to G that we present in this paper are purely group theoretical, even though some parts could also be proved by using characters.…”

“…Q has five maximal abelian subgroups, all of which have order 2 4 . Besides, the product of any two distinct ones of them is the whole of Q.…”

“…(Following [4], we say that G is a Camina group when {[x, g] : g ∈ G} = G for any x ∈ G \ G . Camina groups have been widely studied in the literature: see [4] and the references there. )…”