Tag Archives: presented group

A short tutorial on Tietze transformations

Here is a little tutorial on how to use Tietze transformations. They were named after Austrian mathematician Heinrich Franz Friedrich Tietze. A presentation is a set of generators and relations given by the notation $$\langle~ S~|~W~\rangle$$ where $S$ is set and $W$ is a set of words in the symbols of $S$. $S$ is called […]

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 […]