Generalizing a Rank Analogue of Hall's Theorem

Abstract

Let γ _s (G) and Z _s (G) denote the sth terms of the lower and upper central series of a group G, respectively. In [5] it is shown that if G is a finite nilpotent group and γ _s+1(G) has rank r, then the rank of G/Z _2s (G) is bounded by a function depending only on s and r. In this article we prove that the same result holds under the weaker condition that γ _s+1(G)/γ _s+1(G) ∩ Z _s (G) has rank r. This provides a rank analogue of a generalization of Hall's theorem on finite-by-nilpotent groups.

Key Words:

• Nilpotent groups
• Rank
• Upper and lower central series

2010 Mathematics Subject Classification:

• Primary: 20D15
• 20F14

ACKNOWLEDGMENT

The authors wish to thank the referee for comments and suggestions.

Notes

Communicated by A. Turull.