Homomorphisms: Preserving Subgroups

In the fascinating world of abstract algebra, homomorphisms play a pivotal role. These are functions between groups that respect the group structure, meaning they preserve the operations. When we talk about a homomorphism φ from a group G to a group G' (denoted as φ: G → G'), we're essentially looking at a map that allows us to relate the structure of G to the structure of G'. A fundamental property that makes these mappings so powerful is their ability to preserve subgroups. This means that if you have a subgroup within G, its image under φ will also be a subgroup in G'. Conversely, if you take a subgroup in G', its preimage under φ will be a subgroup in G. Let's delve into these properties to understand why and how homomorphisms act as such reliable preservers of group structure.

Property 1: The Image of a Subgroup is a Subgroup

One of the most significant implications of a homomorphism is its behavior with respect to subgroups. If H is a subgroup of G, then the image of H under φ, denoted as φ[H], is guaranteed to be a subgroup of G'. What does this mean? Well, φ[H] is the set of all elements in G' that are obtained by applying φ to the elements of H. That is, φ[H] = {φ(h) | h ∈ H}. To prove that φ[H] is indeed a subgroup of G', we need to show it satisfies the subgroup criteria: it's non-empty, closed under the group operation, and contains inverses for all its elements. First, since H is a subgroup of G, it must contain the identity element of G, let's call it e. Because φ is a homomorphism, it maps the identity element of G to the identity element of G', so φ(e) = e'. Therefore, e' is in φ[H], ensuring that φ[H] is non-empty. Next, let a' and b' be any two elements in φ[H]. By the definition of φ[H], this means there exist elements a and b in H such that a' = φ(a) and b' = φ(b). Since H is a subgroup, it's closed under the operation, so the product a * b (where '*' denotes the operation in G) is also an element of H. Because φ is a homomorphism, it preserves the group operation: φ(a * b) = φ(a) * φ(b). Substituting our elements, we get φ(a * b) = a' * b'. Since a * b is in H, its image φ(a * b) must be in φ[H]. Thus, a' * b' is in φ[H], proving that φ[H] is closed under the operation in G'. Finally, consider any element a' in φ[H]. This means a' = φ(a) for some a in H. Since H is a subgroup, it contains the inverse of a, denoted as a⁻¹. Because φ is a homomorphism, it preserves inverses: φ(a⁻¹) = (φ(a))⁻¹ = (a')⁻¹. Since a⁻¹ is in H, its image φ(a⁻¹) must be in φ[H]. This shows that the inverse of a' is also in φ[H]. With these three conditions met (non-empty, closed under the operation, and contains inverses), we definitively conclude that φ[H] is a subgroup of G'. This property underscores how homomorphisms effectively transfer the substructure of one group to another, maintaining the essential group properties.

Property 2: The Preimage of a Subgroup is a Subgroup

Complementing the first property, homomorphisms also demonstrate a remarkable ability to