On variants of a theorem of Schur

In a paper [1] published in 2001, we modified a famous theorem of Issai Schur, which asserts that if G is a group with center Z, such that G/Z is finite, then the commutator subgroup G′ = [G,G] is also finite. Our modification was twofold; in the first place, we confined ourselves to nilpotent groups G, so that we could use effective localization methods at an arbitrary family P of primes, and, second, we relativized the situation by replacing G by a pair of groups (G,N), where N is a normal subgroup of G. Then Z was replaced by the centralizer CG(N) of N in G, and [G,G] was replaced by [G,N]. We also considered in [1] a partial converse of Schur’s theorem and its modification. In this partial converse, we showed that G/Z is finite if G′ is finite, provided that G is finitely generated (fg). In talks the author has given on this topic, he has expressed the opinion that the converse would not hold without some supplementary hypothesis. This remark was taken up by Dr. Edwin Clark of the University of South Florida, who raised the question on sci.math.research, and quickly received negative answers from Derek Holt and Andreas Caronti, whose counterexamples were very similar, being infinite extraspecial pgroups. Thus Holt considered a group G with generators xi, yi, i > 0, and z, subject to the relations x p i = y i = zp = 1, [xi,xj] = [yi, yj] = 1, and [xi, yi] = z, [xi, yj] = 1, i = j, and [z,xi] = [z, yi] = 1, for all i. Then Z = G′ = 〈z〉 is finite, but G/Z is infinite. The present author is very grateful to Edwin Clark for providing this elucidation. In Section 2, we provide improved versions of the proofs of the two main theorems of [1], namely, if P is a family of primes with complementary family Q, then Theorems 2.1 and 2.3 hold. It is a striking fact, not brought out in [1], that, whereas the proof of Theorem 2.1 leans heavily on the P-localization theory of nilpotent groups, the proof of Theorem 2.3 does not use localization methods.