Category Archives: group-theory

## A quick intro to Galois descent for schemes

This is a very quick introduction to Galois descent for schemes defined over fields. It is a very special case of faithfully flat descent and other topos-descent theorems, which I won't go into at all. Typically, if you look up descent in an algebraic geometry text you will quickly run into all sorts of diagrams […]

## Explicit example showing non-residual finiteness

This is mostly a continuation on the group I gave in the last post, which is given by the presentation $$G = \langle a,t ~|~ t^{-1}a^2t = a^3\rangle.$$ At the risk of beating a dead horse, I proved that the homomorphism $f:G\to G$ given on generators by $f(t) = t$ and $f(a) = a^2$ is […]

## 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 […]

## 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 […]

## Commutators and the Ore Conjecture

In a talk yesterday by Boris Kunyavski at the University of Ottawa, I learned a little about the Ore conjecture, which in 2010 was proved a theorem in: Liebeck, Martin W.; O'Brien, E. A.; Shalev, Aner; Tiep, Pham Huu. The Ore conjecture. J. Eur. Math. Soc. (JEMS) 12 (2010), no. 4, 939-1008. It's quite a […]

## 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 […]

## Fibonacci sequence modulo m

The Fibonacci sequence is an infinite sequence of integers $f_0,f_1,f_2,\dots$ defined by the initial values $f_0 = f_1 = 1$ and the rule $$f_{n+1} = f_n + f_{n-1}$$ In other words, to get the next term you take the sum of the two previous terms. For example, it starts off with: $$1,1,2,3,5,8,13,21,34,55,\dots$$ You can define […]

## Where does convolution come from?

When I first saw convolution I happily used it without thinking about why we should define it in the first place. Here's a post that might tell you why it makes sense from an algebraic viewpoint. Let's recall the convolution product, as it is for functions $f:\R\to\R$. If $f,g:\R\to\R$ are two such functions, then the […]

## Homomorphisms from G_a to G_m

Let $k$ be a commutative ring. Let $\G_a$ be group functor $\G_a(R) = R$ and $\G_m$ be the group functor $\G_m(R) = R^\times$, both over the base ring $k$. What are the homomorphisms $\G_a\to \G_m$? In other words, what are the characters of $\G_a$? This depends on the ring, of course! The representing Hopf algebra […]