JOURNAL OF ALGEBRA 115, 46-74 (1988) On the Number of Conjugacy Classes in a Finite Group* ANTONIO VERA LOPEZ AND LOURDES ORTIZ DE ELCUEA Departamento de Matembticas, Far&ad de Ciencias, Universidad de1 Pais Yasco, Apartado 644, Bilbao, Spain Communicated by Gernot Stroth Received July 11, 1986 INTRODUCTION In the following, G will denote a finite group and we will use the standard notation of the Theory of Groups. We say that G is a D,-group, if G contains a unique conjugacy class of maximal rc-subgroups. In addition, a D,-group G is said to be a LX-group, if all epimorphic images of sub- groups of G are D,-groups. For example, if G is n-separable, then G is a LX-group. In particular, all rr-solvable groups are L,-groups. In this work, we get new results relative to the conjugacy classes of a finite group G. Let N be a normal subgroup of G, rt a set of prime numbers, r(G) (resp. r”(G)) the number of conjugacy classesof elements (resp. n-elements) of G, rc(G) the set of all different primes dividing IGIl G, the set of all n-elements of G; and G = G/N. In this paper, we analyze the number r”(G) of conjugacy classes of n-elements of G, through the local analysis of the number r;;(gN) of conjugacy classes of n-elements of G which intersect the coset gN. Our aims are threefold: to obtain upper and lower bounds of the number r”(G) of conjugacy classes of rc-elements of G, in terms of the numbers r”(G/N), r”(N), and [G’l, where G’ denotes the derived subgroup of G; to get the residue class of r”(G), modulo the “best” number, given in terms of the primes dividing (GI; and finally, to analyze the conjugacy-vector d, = (IC,(g,)l, . .. . IC,( g,)l) of G, assuming that G is the disjoint union of the classes Cl,(g,), i= 1, . . . . r=r(G), and lCG(g1)3 ... 3 IC,(g,)l. The results obtained are useful both for the calculation of the conjugacy-vector of a finite group and for the classification of finite groups according to the number of conjugacy classes(see Examples 11-13 in Section 4). Moreover, they enable us to obtain the following inequalities: (A) r”(G) < r”(G/N) .r”(N), * This work has been supported by the University of the Basque Country. 46 (1) 0021-8693/88 $3.00 Copyright Q 1988 by Academic Press, Inc. All rights of reproduction in any form reserved.