The basic question that we raise here is the following one: *given a ***group** \(G\) and a **proper subgroup** \(H\) (i.e. \(H \notin \{\{1\},G\}\), can \(G/H\) be isomorphic to \(G\)? A group \(G\) is said to be **hopfian** (after **Heinz Hopf**) if it is not isomorphic with a **proper quotient group**.

All **finite groups** are hopfian as \(|G/H| = |G| \div |H|\). Also, all **simple groups** are hopfian as a simple group doesn’t have proper subgroups.

So we need to turn ourselves to infinite groups to uncover non hopfian groups. Continue reading A (not finitely generated) group isomorphic to a proper quotient group →

## Mathematical exceptions to the rules or intuition