Forcing Linearity Numbers for Abelian Groups

Abstract

Forcing linearity number of an abelian group A is defined as the infimum of cardinalities of sets S of proper subgroups of A such that any homogeneous map f : A → A is an endomorphism of A whenever it is linear on each member of S. The forcing linearity number is determined for all abelian groups. These numbers can be 0, 1, 2, p + 2 (for primes p), or ℵ₀. There is a pathological case: the direct sums of two non-zero cyclic p-groups, where such a number does not exist.

Key Words:

• Abelian group
• Homogeneous map
• Forcing linearity number

Notes

^#Communicated by K. Rangaswamy.