How do you generate a normal subgroup from relations?

Question

From Rotman's Algebraic Topology:

A group $G$ is defined by generators $X = \{x_k \in K\}$ and relations $\Delta = \{r_j = 1 : j \in J\}$ if $G \cong F / R$, where $F$ is the free group on $X$ and $R$ is the normal subgroup of $F$ generated by $\{r_j : j \in J\}$. The ordered pair $(X, \Delta)$ is called a presentation of $G$.

What are the elements of $R$? What does it mean to generate a normal subgroup from a set of relations?

Since I see "generated by $\{r_j : j \in J\}$" I'm assuming it means $\langle r_j : j \in J\rangle$ where $a \in $ iff $a = \sum_{j \in J} r_j^{m_j}$. But since $r_j = 1, \forall j$, then doesn't every element reduce to the identity?

Answer 1

The elements $r_j$ are not equal to $1$ in the free group $F$. But by including them in $R$ they become the identity in the quotient $F/R$.

To avoid confusion in notation, let me use $S$ for the set of relations $r_j$. So $S=\{r_j:j\in J\}$. So the set $S$ is a subset of the free group $F$ on $X$. These are "words" in $X$ that you wish to be the identity in the quotient $F/R$.

In order to have well-defined quotient group, we need $R$ to be:

1. a subgroup

2. normal.

So this is why we take the normal subgroup generated by $S$. By definition this is the subgroup generated by all conjugates $gr_jg^{-1}$ of the relations $r_j$ in $S$. So $$ R=\langle gr_jg^{-1}:j\in J,\space g\in F\rangle $$

Sometimes $S^F$ denotes the set of $F$-conjugates of elements in $S$, and so $R=\langle S^F\rangle$. For example, see https://proofwiki.org/wiki/Definition:Generated_Normal_Subgroup

Answer 2

Given a group $G$ and a subset $S$, you might remember that the subgroup $\langle S \rangle < G$ that generated by $S$ has two equivalent characterizations:

1. $\langle S \rangle$ is the intersection of all subgroups of $G$ that contain $S$, in other words it is the smallest subgroup of $G$ that contains $S$.
2. $\langle S \rangle$ is the subgroup of $G$ consisting of all elements that can be expressed by multiplying out a word, i.e. a finite sequence whose letters are chosen from the set $S \cup S^{-1}$, where $S^{-1} = \{s^{-1} \mid s \in S\}$.

The normal subgroup generated by $S$, which I'll denote $\langle\langle S \rangle\rangle$, has similar characterizations:

1. $\langle\langle S \rangle\rangle$ is the intersection of all normal subgroups of $G$ that contain $S$, in other words it is the smallest normal subgroup of $G$ that contains $S$.
2. $\langle\langle S \rangle\rangle$ is the subgroup of $G$ that is obtained by multiplying out a word whose letters are chosen from the set $\{gsg^{-1} \mid g \in G, s \in S \cup S^{-1}\}$ (in other words, $\langle\langle S \rangle\rangle$ is the subgroup generated by that set).