GROUPS WITH A LARGE CONJUGACY CLASS RELATIVE TO A NORMAL SUBGROUP

In 2013, the first author studied a parameter e defined by the equation vertical bar G vertical bar = root k (root k + e), where k is the largest conjugacy class size of a finite group G. They then obtained a general bound for |G| and studied when equality occurs. Their parameter was motivated by earlier work of Snyder on character degrees; Snyder introduced a new parameter e' and bounded vertical bar G vertical bar in terms of e'. Later, C. Durfee defined a relative version of Snyder's parameter for groups with respect to a fixed normal subgroup N and studied upper bounds for vertical bar G/N vertical bar in terms of the relative parameter. Following the lead on the character theoretic side again, we define the conjugacy class analogue eN of the relative parameter; in particular, we obtain the tight upper bound vertical bar G/N vertical bar <= p/(p - 1)(2) (e(N))(2) and study when equality occurs. In addition, we establish some interesting properties for e(N).