CONJUGACY CLASSES IN FINITE p-GROUPS

Of course, in that problem we have to take into account that the class sizes impose restrictions on the group structure. E.g. if the sizes are {1, p}, then the nilpotency class has to be 2. More precisely: the class sizes of a p-group G are {1, p} iff |G′| = p (Knoche; see also Theorem 3 below). But we can ask, e.g., if, given any set S ≠ {1, p} of p-powers, does there exist a group of class 3 whose class sizes are the members of S. Given an element x ∈ G whose class size is p, we say that b = b(x) is the breadth of x. The breadth b(G) of G is the maximal breadth of its elements. There is much interest in the relation of this invariant to other invariants of G which measure its departure from commutativity. The following is obvious.