We consider a **vector space** \(E\) and a **linear map** \(T \in \mathcal{L}(E)\) having a **left inverse** \(S\) which means that \(S \circ T = S T =I\) where \(I\) is the **identity map** in \(E\).

When \(E\) is of finite dimension, \(S\) is **invertible**. Continue reading A linear map having a left inverse which is not a right inverse →

We take a **metric space** \((E,d)\) and consider two **closed subsets** \(A,B\) having a **distance** \(d(A,B)\) equal to zero. We raise the following question: *can \(A\) and \(B\) be disjoint – \(A \cap B=\emptyset\)?* Continue reading Two disjoint closed sets with distance equal to zero →

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 →

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