van Kampen’s Theorem

Abstract

The notation for this chapter will be as follows: if $G$ is a group and $S\subset G$ a subset we will write $\langle S\rangle \subset G$ or $\langle s\mid s\in S\rangle \subset G$ for the normal subgroup generated by $S$, which is the intersection of all normal subgroups in $G$ containing $S$. It’s easy to show that $\langle S\rangle$ coincides with the subgroup generated by all elements of the type $gsg^{-1}$, for $s\in S$ and $g\in G$.