# Isomorphism of Cyclic Groups

__Theorem 1__**:**

Cyclic groups of the same order are isomorphic.

__Proof__**: **Let $$G$$ and$$\,G’$$ be two cyclic groups of order $$n$$, which are generated by $$a$$ and $$b$$ respectively. Then

\[G = \left\{ {a,{a^2},{a^3}, \ldots ,{a^n} = e} \right\}\] and \[G’ = \left\{ {b,{b^2},{b^3}, \ldots ,{b^n} = e’} \right\}\]

The mapping $$f:G \to G’$$, defined by $$f\left( {{a^r}} \right) = {b^r}$$, is isomorphism.

For, \[f\left( {{a^r} \cdot {a^s}} \right) = f\left( {{a^{r + s}}} \right) = {b^{r + s}} = {b^r} \cdot {b^s} = f\left( {{a^r}} \right) \cdot f\left( {{a^s}} \right)\]

Therefore the groups are isomorphic.

__Theorem 2__**: **

An infinite cyclic group is isomorphic to the additive group of integers.

__Proof__**:** Let $$G$$ be an infinite cyclic group, generated by $$a$$, then

\[G = \left\{ { \ldots ,{a^{ – 2}},{a^{ – 1}},{a^0} = e,{a^1},{a^2},{a^3}, \ldots } \right\} = \left\{ {{a^r}:r\,{\text{is}}\,{\text{an}}\,{\text{integer}}} \right\}\]

The mapping $$f:G \to \mathbb{Z}$$, defined by $$f\left( {{a^r}} \right) = r$$ is an isomorphism, for it is one-one onto, and further,

\[\begin{gathered} f\left( {{a^r} \cdot {a^s}} \right) = f\left( {{a^{r + s}}} \right) = r + s \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = f\left( {{a^r}} \right) + f\left( {{a^s}} \right) \\ \end{gathered} \]

It follows that $$G$$ is isomorphic to $$\mathbb{Z}$$.

__Theorem 3__**:**

A cyclic group of order $$n$$ is isomorphic to the additive group of residue classes modulo $$n$$.

__Proof__**:** Let $$G$$ be an infinite cyclic group, generated by $$a$$, then

\[G = \left\{ {a,{a^2},{a^3}, \ldots ,{a^{n – 1}},{a^n} = e} \right\}\]

Let $$G’$$ be the additive group or residue classes $$\left( {\bmod n} \right)$$, i.e.

\[G’ = \left\{ {\left[ 1 \right],\left[ 2 \right],\left[ 3 \right], \ldots ,\left[ n \right] = \left[ 0 \right]} \right\}\]

The mapping $$f:G \to G’$$, defined by $$f\left( {{a^r}} \right) = \left[ r \right]$$, is isomorphism, for it is one-one onto, and further,

\[\begin{gathered} f\left( {{a^r} \cdot {a^s}} \right) = f\left( {{a^{r + s}}} \right) = \left[ {r + s} \right] = \left[ r \right] + \left[ s \right] \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = f\left( {{a^r}} \right) + f\left( {{a^s}} \right) \\ \end{gathered} \]

It follows that $$G$$ is isomorphic to $$G’$$.

__Theorem 4__**:**

A subgroup of the infinite cyclic group is isomorphic to the additive group of integral multiples of an integer.

__Proof__**:** Let $$G = \left\{ { \ldots ,{a^{ – 2}},{a^{ – 1}},{a^0} = e,{a^1},{a^2},{a^3}, \ldots } \right\}$$ and let $$H$$ be a subgroup of $$G$$, given by,

\[H = \left\{ { \ldots ,{a^{ – 2m}},{a^{ – m}},{a^0} = e,{a^m},{a^{2m}}, \ldots } \right\} = \left\{ {{{\left( {{a^m}} \right)}^n}:n \in \mathbb{Z}} \right\}\]

Then $$H$$ is isomorphic to the additive group $$H’$$, given by

\[H’ = \left\{ {0, \pm m, \pm 2m, \pm 3m, \ldots } \right\} = \left\{ {nm:n \in \mathbb{Z}} \right\}\]

The mapping $$f:H \to H’$$, defined by $$f\left( {{a^{mn}}} \right) = nm$$, is isomorphism, for it is one-one onto, and if $$r,s \in \mathbb{Z}$$, then

\[\begin{gathered} f\left( {{a^{rm}} \cdot {a^{sm}}} \right) = f\left( {{a^{\left( {r + s} \right)m}}} \right) = {\left( {r + s} \right)^m} = rm + sm \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = f\left( {{a^{rm}}} \right) + f\left( {{a^{sm}}} \right) \\ \end{gathered} \]

It will be observed that $$H$$ is itself an infinite cyclic group, and as such it is isomorphic to $$G$$. Thus a subgroup of an infinite cyclic group is isomorphic to the group itself.