On a theorem of Schur

Abstract

We study the ramifications of Schur's theorem that, ifGis a group such thatG/ZGis finite, thenG′is finite, if we restrict attention to nilpotent group. HereZGis the center ofG, andG′is the commutator subgroup. We use localization methods and obtain relativized versions of the main theorems.

The converse of Schur’s theorem

“…Hilton in [1] proved the following theorem Theorem 1.1. If G is a finitely generated group such that G is finite, then G/Z(G) is finite.…”

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 C G (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 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.…”