MA THEMA TICS ON n-ISOCLINIC GROUPS BY J. C. BIOCH (Communioated by Prof. J. P. MURRE at the meeting of June 19, 1976) The notion of isoclinism was introduced by P. Hall [2]. In [1] we have proved that monomiality is an invariant of the families of finite isoclinic groups. In this paper we consider a more general form of isoclinism, called n-isoclinism, and we prove that strong-monomia1ity is a family- invariant for finite n-isoclinic groups. Moreover, using a theorem of P. M. Weichsel [7] we give short proofs for results of P. Hall [2] and J. Tappe [6] on the irreducible characters of isoclinic groups. As a corollary we obtain the above mentioned result on the M -group property proved in [1]. Notations are standard and can be found in Huppert's book [4]. AOKNOWLEDGEMENT I express my gratitude to Dr. J. Tappe, who drew my attention to a result of P. M. Weichsel on isoclinic groups. 1. n-ISoCLINIO GROUPS The notion of n-isoclinism of groups is implicit in a short note of P. Hall [3] on verbal and marginal subgroups. Let G=K I (G»K 2 (G» ... be the lower central series of the group G. Each term of this series, being generated by commutator words, is a verbal subgroup. An element g of Gis ca.lled a period of Kn(G), if for all simple commutators [g1, ... , gn] E Kn(G) we have [g1, .. ·,glg, ... ,gn]=[gl, ... ,gl, ... ,gn], j=l, 2, ... ,n. The set of all periods of a verbal subgroup X is called the marginal subgroup of X. The marginal subgroup of K,(G) is Z'-I(G), where the latter group is the (i-l)-th term of the upper central series of G: 1 =Zo(G)<Zl(G)=Z(G)<Z2(G) < .... As is well-known, the subgroups K,(G) and Z,(G) centralize each other, see [4] theorem 111.2.11. 1.1. DEll'INITION. Two groups G and H are n-isoclinic, G'7' H, if there exist isomorphisms 0(, and {J: 0(,: G/Z,.(G) H/Zn(H)