Let \(\mathbb{Q}_2\) be the **ring** of rational numbers of the form \(m2^n\) with \(m, n \in \mathbb{Z}\) and \(N = U(3, \mathbb{Q}_2)\) the group of **unitriangular matrices** of dimension \(3\) over \(\mathbb{Q}_2\). Let \(t\) be the **diagonal matrix** with diagonal entries: \(1, 2, 1\) and put \(H = \langle t, N \rangle\). We will prove that \(H\) is **finitely generated** and that one of its **quotient group** \(G\) is **isomorphic** to a proper quotient group of \(G\). Continue reading A finitely generated soluble group isomorphic to a proper quotient group

# Tag Archives: hopfian

# A (not finitely generated) group isomorphic to a proper quotient group

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