ABSTRACT
Preview Activity 30.1. As we saw in Investigation 29, the notion of isomorphism formalizes what
it means for two groups to be essentially the same. Recall that an isomorphism of groups is a
bijective, structure-preserving function. In group theory, structure-preserving maps are important
even if they are not bijections. In this activity, we will explore three different kinds of structure-
preserving functions. (Throughout the activity, recall that we use the notation [k]n to denote the congruence class of k in Zn.)
(a) Is the function ϕ : Z3 → Z6 defined by ϕ([k]3) = [4k]6 structure-preserving? Is ϕ an injection? Is ϕ a surjection? Verify your answers. (You may assume that ϕ is well-defined.)
(b) Is the function ϕ : Z6 → Z3 defined by ϕ([k]6) = [k]3 structure-preserving? Is ϕ an injection? Is ϕ a surjection? Verify your answers. (You may assume that ϕ is well-defined.)
(c) Is the function ϕ : Z6 → Z4 defined by ϕ([k]6) = [2k]4 structure-preserving? Is ϕ an injection? Is ϕ a surjection? Verify your answers. (You may assume that ϕ is well-defined.)
Preview Activity 30.1 illustrates that it is possible to have structure-preserving maps that are injec-
tions but not surjections, surjections but not injections, or neither surjections nor injections. When
we study groups, we are mostly interested in maps that preserve the group structure or operation.
Such maps-whether they are injective, surjective, neither, or both-are called homomorphisms,
defined formally as follows:
