Abstract Algebra/Group Theory/Meaning of Diagrams in This Section

If the formal is rigorous but obscures ideas behind lines after lines of proofs, what can the visual offer?

In our mind, we all have the experience that logic to a degree can depend only on image without symbols or language.

Below is an attempt to replace the formal proofs with equally rigorous diagrams.

Your suggestions in the discussion of this page is much appreciated. The problems in this attempt will also be in the discussion.