A1-MilaSchultz

Good Visualization

Source: Algebra by Michael Artin (1991), page 56

This diagram appears in an Algebra textbook for upper-level undergraduate and graduate students. It's main purpose is to help students create a mental model of group homomorphisms. The diagram actually contains a significant amount of meaningful information while appearing simple and uncluttered. I included part of the text of the book to demonstrate how the worlds alone may not necessarily give students a helpful mental model of the structures. The diagram clarifies how homomorphisms act upon groups; it is apparent that the structure on the left maps to the structure on the right under the action of the homomorphism. The diagram is extremely helpful in allowing students to mentally manipulate homomorphisms of groups, more than an entire section of material without overloading the diagram with specifics.

In a nutshell, the diagram shows that small distinct subsets of G are mapped to one element each in G'. With a few extra small details, however, the diagram tells much more without obfuscating the main point.

Some information in the diagram:

• The homomorphism goes from G to G'
• G and G' need not be the same shape (size)
• The kernel of the homomorphism a subset of G
• The kernel of the homomorphism is often written as N
• The identity in G, 1, is in the kernel of the homomorphism
• An element a in G is multiplied by N, which is visually the same as N, but centered around a rather than the 1, the identity. This is accurate because N and aN have the same number of elements, and 1 and a play respectively similar roles.
• The entire group G is mapped to a subset of G' which is not necessarily the same size or shape as G, called the image of the homomorphism.
• The kernel maps to the identity in G', which is in the image of the homomorphism.
• The entire subset aN maps to the element that a itself is mapped to in G'
• N and aN are mapped similarly, and both seem to "collapse" into one element each in G'.
• The elements of G' in the image of the homomorphism are of the form φ(a).