# Calculating a presentation of $\mathbb{Z}_{3}$ in detail.

Theorem: Let $$G$$ groups and $$S\subset G$$ such that $$\langle S\rangle =G$$. (Here $$G=\left\{s_1\ldots, s_n:s_i\in S\cup S^{-1}, n\in\mathbb{N}\right\}$$.) Let $$\varphi:S\to G$$ with $$\varphi(s)=s$$. By the universal property of free groups there exists a unique homomorphism (in fact, epimorphism) $$\varphi:F(S)\to G$$ with $$F(S)=\left\{w\in S^{\ast}: w \text{ reduced word} \right\}.$$Then $$G\simeq F(S)/{ker(\varphi)}.$$

Here $$\langle \langle S\rangle\rangle=\langle \left\{gsg^{-1}:s\in S\cup S^{-1},\ g\in G\right\}\rangle.$$

Let $$S\subset G$$ and $$G=\langle S\rangle.$$ Then $$\langle S\mid T\rangle$$ presentation of $$G$$ if $$G=\langle S\rangle$$ and $$T\subset \ker\varphi$$ and $$\langle \langle T\rangle\rangle=\ker\varphi$$.

I want prove in a detailed way that $$\mathbb{Z}_{3}=\langle a\mid a^3\rangle.$$

## I have this.

Here $$S=\left\{a\right\}.$$

First, $$a$$ must be different from $$0$$. Because if $$a=0$$, then $$\mathbb{Z}_{3}=\left\{0\right\}$$.

If $$a=1,$$ then $$0=1+1+1, 1=1, 2=1+1$$.

If $$a=2$$, then $$0=2+2+2, 1=2+2^{-1}, 2=2$$.

Therefore, $$\mathbb{Z}_{3}=\langle a\rangle$$, with $$a=1$$ or $$a=2$$.

So . . .

How prove $$\ker(\varphi)=\langle\langle \left\{a^3\right\}\rangle\rangle$$?

## I have this:

$$\ker\varphi=\left\{s_1\cdots s_n\in F(S): \varphi(s_1\cdots s_n)=0,\ s_i\in \left\{a\right\}\cup\left\{a^{-1}\right\}, n\in\mathbb{N}\right\}$$

$$=\left\{s_1\cdots s_n\in F(S): s_1+\cdots +s_n=0,\ s_i\in \left\{a\right\}\cup\left\{a^{-1}\right\}, n\in\mathbb{N}\right\}$$

• There are many different ways of defining $\Bbb Z_3$ and a sufficiently detailed answer would depend on which definition you are using. Please be more specific in future. – Shaun Dec 14 '18 at 3:32
• Note, too, that there is no such thing as the presentation of a group, strictly speaking. – Shaun Dec 14 '18 at 4:36

In order to show $$\ker(\varphi)=\langle \langle a^3\rangle\rangle$$, one must first be clear on what $$\varphi$$ is, and instead of writing, say, $$\color{red}{a=1}$$, one writes $$\varphi(a)=1$$.

You are on the right track in that you have evidence to suggest that $$\varphi: a\mapsto 1\text{ or } 2$$; that is, that the generator $$a$$ of the presentation $$\langle a\mid a^3\rangle$$ maps via $$\varphi$$ to one of the elements of $$\Bbb Z_3$$ given by a number coprime to $$3$$.

What reduced words, with letters in $$\{ a, a^{-1}\}$$, get mapped to the identity of $$\Bbb Z_3$$ via $$\varphi$$?
But can you deduce what each word maps to under $$\varphi$$ in general?
Hint: Use Bézout's Identity, assuming $$\varphi(a)=p$$ for some $$p$$ coprime to $$3$$. Here $$\varphi$$ is a homomorphism. Don't forget to show both $$\ker(\varphi)\subseteq \langle\langle a^3\rangle\rangle$$ and $$\langle\langle a^3\rangle\rangle\subseteq\ker(\varphi)$$.