Basic Theorems Regarding Cyclic Groups

Basic Theorems Regarding Cyclic Groups

Recall from the Cyclic Groups page that if $G$ is a group then $G$ is said to be cyclic if there exists an $a \in G$ such that $G = \langle a \rangle$, that is, every element of $G$ is of the form $a^n$ for some $n \in \mathbb{Z}$. We will now look at some basic results regarding cyclic groups.

 Theorem 1: If $G$ is an infinite cyclic group then $G$ is isomorphic to $(\mathbb{Z}, +)$.
• Proof: Let $G$ be an infinite cyclic group and write $G = \langle a \rangle$. If $g \in G$ then there exists a unique $n \in \mathbb{Z}$ such that $g = a^n$. Immediately, we see that $|G|$ is countably infinite.
• Let $\varphi : G \to \mathbb{Z}$ be defined for each $g = a^{n(g)} \in G$ by:
(1)
\begin{align} \quad \varphi(g) = \varphi(a^{g(n)}) := n(g) \end{align}
• Then $\varphi$ is a homomorphism from $G = \langle a \rangle$ to $(\mathbb{Z}, +)$ since for all $g_1 = a^{n(g_1)}, g_2 = a^{n(g_2)} \in G$ we have by the laws of exponents that:
(2)
\begin{align} \quad \varphi(g_1 \cdot g_2) = \varphi(a^{n(g_1)}a^{n(g_2)}) = \varphi(a^{n(g_1) + n(g_2)}) = n(g_1) + n(g_2) = \varphi(g_1) + \varphi(g_2) \end{align}
• Let $g_1 = a^{n(g_1)}, g_2 = a^{n(g_2)} \in G$ and suppose that $\varphi(g_1) = \varphi(g_2)$}]. Then [[$n(g_1) = n(g_2)$, so $g_1 = a^{n(g_1}) = a^{n(g_2)} = g_2$, so $\varphi$ is injective.
• Furthermore, if $n \in \mathbb{Z}$ then $a^{n}$ is such that $\varphi(g^n) = n$, so $\varphi$ is surjective.
• Thus $\varphi : G \to \mathbb{Z}$ is an isomorphism of $G = \langle a \rangle$ to $(\mathbb{Z}, +)$. $\blacksquare$
 Theorem 2: If $G$ is a finite cyclic group of order $n$ then $G$ is isomorphic to $(\mathbb{Z}/n\mathbb{Z}, +)$.
• Proof: Suppose that $G = \langle a \rangle$. Then $G = \{ a^0, a^1, ..., a^{n-1} \}$. Let $\varphi : G \to \mathbb{Z}/n\mathbb{Z}$ be defined by:
(3)
\begin{align} \quad \varphi(a^i) = [i] \end{align}
• Then $\varphi$ is a homomorphism since if $a^i, a^j \in G$ is such that $i + j \leq n-1$ then:
(4)
\begin{align} \quad \varphi(a^i \cdot a^j) = \varphi (a^{i+j}) = [i] + [j]= \varphi(a^i) + \varphi(a^j) \end{align}
• And if $a^i, a^j \in G$ is such that $i + j > n-1$ then we can write $i + j = qn + r$ where $q \geq 1$ and $0 \leq r < n$ (by the division algorithm) so that:
(5)
\begin{align} \quad \varphi(a^i \cdot a^j) = \varphi(a^{i+j}) = \varphi(a^{qn + r}) = \varphi((a^n)^qa^r) = \varphi(1^qa^r) = \varphi(a^r) = [r] = [qn + r] = [i +j] \end{align}
• It is easy to see that $\varphi$ is injective and surjective, so $\varphi$ is an isomorphism of $G = \langle a \rangle$ and $(\mathbb{Z}/n\mathbb{Z}, +)$. $\blacksquare$
 Proposition 3: Two finite cyclic groups $G_1$ and $G_2$ are isomorphic if and only if they are of the same order.
 Proposition 4: If $G$ is a cyclic group and $H$ is a subgroup of $G$ then $H$ is also cyclic group. Furthermore, if $G$ is a finite cyclic group of order $n$ then for each $k \in \mathbb{N}$ with $k | n$ there exists a unique subgroup $H$ of $G$ with $|H| = k$.
 Proposition 5: Let $G$ be a finite cyclic group of order $n$ with $G = \langle a \rangle$. Then $G$ is generated by $a^k$ if and only if $\mathrm{gcd}(n, k) = 1$.