Category Archives: math

Anything mathematical.

Dividing a square into triangles of equal area

Take a square and divide it down a diagonal, dividing the square into two triangles. Drawing the opposite diagonal now divides it into four triangles. In these two examples, we divided a square into an even number of triangles, all with equal area. Can we divide a square into an odd number of nonoverlapping triangles, […]




Yet another group that is not Hopfian

A few weeks ago I gave an example of a non-Hopfian finitely-presented group. Recall that a group $G$ is said to be Hopfian if every surjective group homomorphism $G\to G$ is actually an isomorphism. All finitely-generated, residually finite groups are Hopfian. So for example, the group of the integers $\Z$ is Hopfian. Another example of […]



A finitely generated flat module that is not projective

Let's see an example of a finitely-generated flat module that is not projective! What does this provide a counterexample to? If $R$ is a ring that is either right Noetherian or a local ring (that is, has a unique maximal right ideal or equivalently, a unique maximal left ideal), then every finitely-generated flat right $R$-module […]


Kourkovka Notebook: Open problems in group theory

Every once in a while I spot a true gem on the arXiv. Unsolved Problems in Group Theory: The Kourkovka Notebook is such a gem: it is a huge collection of open problems in group theory. Started in 1965, this 19th volume contains hundreds of problems posed by mathematicians around the world. Additionally, problems solved […]


Britton's lemma and a non-Hopfian fp group

In a recent post on residually finite groups, I talked a bit about Hopfian groups. A group $G$ is Hopfian if every surjective group homomorphism $G\to G$ is an isomorphism. This concept connected back to residually finite groups because if a group $G$ is residually finite and finitely generated, then it is Hopfian. A free […]



What is a residually finite group?

We say that a group $G$ is residually finite if for each $g\in G$ that is not equal to the identity of $G$, there exists a finite group $F$ and a group homomorphism $$\varphi:G\to F$$ such that $\varphi(g)$ is not the identity of $F$. The definition does not change if we require that $\varphi$ be […]