Exercise 1.2.15 (Set of generators and relations for $\mathbb{Z}/n\mathbb{Z}$)

Question

Find a set of generators and relations for $\mathbb{Z}/n\mathbb{Z}$.

Richard Ganaye · Accepted Answer

ewcommand{\Z}{\mathbb{Z}}

We replace $\mathbb{Z}/n\mathbb{Z}$ by the isomorphic cyclic group $C_n$, with multiplicative notations.

We show that $C_n = \langle a \mid a^n = e angle$, so the set of generator is $S = \{a\}$ and the set of relations is $R = \{a^n\}$.

(Since the The definition given in section 1.2 is rather vague, we use the more precise definition of section 6.3.)

This may seem curious at first glance, since we only express $a^n = e$, but not that $a$ is of order $n$, i.e. $a^{d} 
e e$ if $d\mid n, d 
e n$.

\begin{proof}

To explain this presentation, it suffices to note that by definition of $C_n$, $C_n$ is generated by an element $a$ of order $n$. The free group on $S = \{a\}$ is the infinite monogeneric group $L  = L(a) \simeq \Z$. Then $L$ is abelian, and the smallest (normal) subgroup containing $a^n$ is $\langle a^n angle = \{a^k, \ n \mid k\}$ isomorphic to $n\Z$. Therefore $L/N \simeq \Z/n\Z \simeq C_n$. By section 6.3, we obtain
$$C_n \simeq L/N \simeq \langle a \mid a^n angle.$$
\end{proof}

Note: This presentation implies that there exists a unique homomorphism from $C_n$ to any group $H$ satisfying $\alpha^n = e$ for some $\alpha \in H$.

Exercise 1.2.15 (Set of generators and relations for $\mathbb{Z}/n\mathbb{Z}$)

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Exercise 1.2.15 (Set of generators and relations for $\mathbb{Z}/n\mathbb{Z}$)

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