# Extensions of Homomorphisms

Chapter

## Abstract

Let (*X*, +) be a group, and (*S*, +) a subsemigroup (i.e., a semigroup such that *S* ⊂ *G* and the operation + is the same as in *X*; cf. 4.5). Let (*Y*, +) ba another group, and let *g* : *S* → *Y* be a homomorphism^{1}. The problem with which we shall deal in 18.1–18.4 is the following. Does there exist a homomorphism f : *X* → *Y* such that *f* Ç *S* = *g*? The main result in this section (cf. Dhombres-Ger [70], Balcerzyk [18]) reads as follows.

## Keywords

Commutative Group Convex Function Normal Subgroup Arbitrary Function Additive Function
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