Isomorphism of Cyclic Groups

Theorem 1:
Cyclic group of 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

and

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

For,

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 groups, generated by $a$, then

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

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 groups, generated by $a$, then

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

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,

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 $G$, give by,

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

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

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.