Kernel of Homomorphism on Cyclic Group

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Theorem

Let $G = \gen g$ be a cyclic group with generator $g$.

Let $H$ be a group.

Let $\phi: G \to H$ be a (group) homomorphism.

Let $\map \ker \phi$ denote the kernel of $\phi$.

Let $\Img G$ denote the homomorphic image of $G$ under $\phi$.


Then:

$\map \ker \phi = \gen {g^m}$

where:

$m = 0$ if $\Img \phi$ is an infinite cyclic group
$m = \order {\Img \phi}$ if $\Img \phi$ is a finite cyclic group.


Proof

From Kernel of Group Homomorphism is Subgroup and Subgroup of Cyclic Group is Cyclic:

$\exists m \in \N: \map \ker \phi = \gen {g^m}$

From Homomorphic Image of Cyclic Group is Cyclic Group:

$\Img \phi$ is a cyclic group generated by $\map \phi g$.


Case $1$: $\Img \phi$ is infinite

Aiming for a contradiction, suppose $m \ne 0$.

Then $g^m$ is not the identity.


Thus:

\(\ds \map \phi {g^m}\) \(=\) \(\ds \paren {\map \phi g}^m\) Homomorphism of Power of Group Element
\(\ds \) \(\ne\) \(\ds e_H\) Definition of Infinite Cyclic Group

However this contradicts $g^m \in \gen {g^m} = \map \ker \phi$.

Hence we must have $m = 0$.

$\Box$


Case $2$: $\Img \phi$ is finite

From Order of Cyclic Group equals Order of Generator, the order of $\map \phi g$ is $\order {\Img \phi}$.

Let $n \in \Z$.

By Division Theorem:

$\exists q, r \in \Z: 0 \le r < \order {\Img \phi}: n = q \order {\Img \phi} + r$

We have:

\(\ds \map \phi {g^n}\) \(=\) \(\ds \paren {\map \phi g}^n\) Homomorphism of Power of Group Element
\(\ds \) \(=\) \(\ds \paren {\map \phi g}^{q \order {\Img \phi} + r}\)
\(\ds \) \(=\) \(\ds \paren {\paren {\map \phi g}^{\order {\Img \phi} } }^q \paren {\map \phi g}^r\)
\(\ds \) \(=\) \(\ds {e_H}^q \paren {\map \phi g}^r\) as $\paren {\map \phi g}^{\order {\Img \phi} } = e_H$
\(\ds \) \(=\) \(\ds \paren {\map \phi g}^r\) Power of Identity is Identity

From definition of order of group element:

$0 < r < \order {\Img \phi} \implies \paren {\map \phi g}^r \ne e_H$

Hence:

\(\ds \map \phi {g^n} = e_H\) \(\iff\) \(\ds r = 0\)
\(\ds \leadstoandfrom \ \ \) \(\ds g^n \in \ker \phi\) \(\iff\) \(\ds n = q \order {\Img \phi}\) Definition of Kernel of Group Homomorphism
\(\ds \leadstoandfrom \ \ \) \(\ds g^n \in \gen {g^m}\) \(\iff\) \(\ds g^n \in \gen {g^\order {\Img \phi} }\) Group Generated by Singleton
\(\ds \leadstoandfrom \ \ \) \(\ds \gen {g^m}\) \(=\) \(\ds \gen {g^\order {\Img \phi} }\) Definition of Set Equality

and we see that $m = \order {\Img \phi}$ satisfies the above relation.

$\blacksquare$


Sources

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