What does the author want to say about generators and relations in group theory? ("Algebra 1st Edition" by Michael Artin)

Question

I am reading "Algebra 1st Edition" by Michael Artin.
I want to know about generators and relations because I think I need to know about generators and relations when I use the GAP software.
But generators and relations are abstract and very difficult for me.

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1. The author wrote "$\phi(w)=1$ or $w=1$ in $G$".
   Do we need "or $w=1$ in $G$"?
   I cannot understand what the author wants to say.

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2. The author wrote:

It might seem more systematic to require the defining relations to be generators for the group $N$. But remember that the kernel of the homomorphism $F\to G$ defined by a set of generators is always a normal subgroup, so there is no need to make the list of defining relations longer. If we know that some relation $r=1$ holds in $G$, then we can conclude that $grg^{-1}=1$ holds in $G$ too, simply by multiplying both sides of the equation on the left and right by $g$ and $g^{-1}$.

I cannot understand what the author wants to say in the above sentences.

Please explain what the author wants to say in the above sentences if possible.

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Thank you.

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Answer 1

Regarding your question 1, the sentence

$\phi(w)=1$ or $w=1$ in $G$

is an explanation of an alternate notation, a notational abuse that is common in group theory. You can read this sentence like this:

$\phi(w)=1$ or [by abusing notation and regarding the letters in the word $w$ as actual elements of $G$, and then multiplying those elements together in the order given using the group operation of $G$] $w=1$ in $G$.

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Regarding your question 2, the author is explaining what information one needs in order to completely specify a normal subgroup $N < F$ (and then the author applies that explanation that to the case that $N$ is the kernel of the homomorphism $\phi : F \to G$).

For general subgroups, not assumed to be normal, one usually specifies a subgroup by listing a finite generating set for the subgroup. [It might seem more systematic to require the defining relators to be generators for the group $N$].

However, for a normal subgroup $N$, such as the kernel of the homomorphism $\phi : F \to G$, one does not need to list an entire generating set in order to determine $N$. [But remember that the kernel of the homomorphism $F \mapsto G$ defined by a set of generators is always a normal subgroup, so there is no need to make the list of defining relators longer].

One can instead list (what is usually) a much smaller set of elements $\{r_j \mid j \in J\} \subset N$ having the property that their set of conjugates $\{g r_j g^{-1} \mid j \in J, g \in G\}$ generates $N$. [If we know that some relation $r=1$ holds in $G$, then we can conclude that $grg^{-1}=1$ holds in $G$ too, simply by multiplying both sides of the equation on the left and right by $g$ and $g^{-1}$]

To summarize, one says that the set $\{r_j \mid j \in J\}$ normally generates $N$ if its set of conjugates generates $N$ in the ordinary sense. This is equivalent to saying that $N$ is the smallest normal subgroup in $G$ that contains the set $\{r_j \mid j \in J\}$, which is also equivalent to saying that $N$ is the intersection of all normal subgroups of $G$ that contain the set $\{r_j \mid j \in J\}$. So, when $N$ is the kernel of the homomorphism $F \mapsto G$, the condition on a set of relators in a presentation for $G$ is that the set normally generates $N$.