Groups with No Nontrivial Linear Representations: Corrigenda

Professor B.A.F. Wehrfritz has kindly drawn my attention to some errors in the above paper [1]. Principally, there is an invalid assertion of complete irreducibility used in Theorem 2.1 and its corollary. Thus the expression " 1-dimensional f-representation" should be replaced by "Abelian 6-representation" in both statements. This amended conclusion is still adequate for subsequent results. In particular, the "if" direction of Theorem 2.3 (a) may be proved as follows. One needs the existence of a nontrivial locally finite quotient. When G is perfect, this is immediate from the stated fact that G is solubleby-locally finite. Otherwise, G admits a nontrivial Abelian, counter-rational quotient, which, when embedded in a divisible group, again leads to a nontrivial locally finite quotient. Likewise, the statement of Corollary 2.4 is unaffected: (ii) implies (iii) from (2.3)(/9) and the proof of (2.3)(b). In Proposition A.I, (ii) should be amended to read either "C C (counter) — C if C is closed under the formation of quotients" or "C C (counter) — C only if C is closed under the formation of simple quotients", while (iv) should be deleted. Incidentally, another derivation of (v) may be found in [3], where it is also noted that the class of counterC groups is always extension-closed if C is closed under the formation of quotients. (In fact, C C (counter) — C suffices.) The class of counter-finite groups also arise in a significant way in [2]. Example 1.9 cannot be sustained, because a linear group with unipotent generators need not be unipotent. However, the relationship between strongly torsion generated groups and counter-linearity seems worth investigating.