Monomiality and isoclinism of groups* By Jan Corstiaan Bioch at Rotterdam and Robert Willem van der Waall at Nijmegen Introduction In [5] P. Hall introduces a principle of classification of solvable groups (in particular öf prime-power groups) that is based on some special equivalence relation on groups, called isoclinism. This relation yields a classification of all groups into mutually exclusive classes of groups, such that all abelian groups collapse: all abelian groups are equivalentto 1. Two groups G i and G 2 in the same equivalence-class (isoclinism-dass) are called isoclinic: G 1 ^G 2 . Roughly speaking two groups are isoclinic if and only if there exists an isomorphism between their central quotients which induces an isomorphism between their commutator subgroups, see definition 1.1. In this part we prove that monomiality is a class-invariant, in Hall's sense 1 ), if we restrict ourselves to finite groups. The same 2 ) can be proved for strong-monomiality 3 ), and in a trivial way for nilpotency and supersolvability. This paper is organized äs follows: In § l we mention some easy consequences of the definition of isoclinism. In § 2 the following Situation will be considered: Let G be a finite group such that G~H 9 where H is resp. a subgroup or factor group of G. In this case the non-linear irreducible characters of G can be obtained from the irreducible characters of H. In § 3 we deal with representation groups (Schur's Darstellungsgruppen) of a finite group G. These groups play an important role in the theory of I. Schur on projective representations, see Schur [10]. It is known that two representation groups G t and G 2 of a finite group H are isoclinic, see Gruenberg [4] section 9. 9, th. 7. We prove the following stronger result: there exists a finite group G such that G l ~G~G 2 and such that *) The results of this paper were announced at the meeting in Oberwolfach on group theory, August 1974. *) Cf. [5] p. 133: "Any quantity depending on a variable group and which is the same for any two groups of the same family will be called a family invariant", 2 ) Even in the case in which we consider the more general concept of weak-isoclinism, see § 5. 3 ) A finite group G will be called strongly-monomial if G and all its subgroups are monomial. Brought to you by | University of Iowa Libraries Authenticated Download Date | 6/8/15 10:36 PM