Therefore $gH=(aH)^i$ for any kind of coset $gH$.so $G/H$ is cyclic , by meaning of cyclic groups.

You are watching: Prove that a factor group of a cyclic group is cyclic

How $gH=(aH)^i$ of any type of coset $gH$.

proves variable group to be cyclic.

Please explain.



The discussion shows the each $gH\in G/H$ is a strength of the single fixed element $aH$. In other words, $G/H=\(aH)^i:i\in\Bbb Z\=\langle aH\rangle$: the cyclic subgroup the $G/H$ generated by $aH$ is the whole group $G/H$, i beg your pardon is thus cyclic.


Here is a slightly different proof that ns hope will certainly clarify things. A team $G$ is cyclic if, and also only if, there is a surjective homomorphism $\thedailysplash.tvbb Z \to G$. Now, consider any factor team $G/H$. Climate there is the canonical surjection $G \to G/H$. Now, if $G$ is cyclic climate there is a surjective homomorphism $\thedailysplash.tvbb Z\to G$. The composite $\thedailysplash.tvbb Z\to G\to G/H$ is then a surjective homomorphism (since the composite that surjections is a surjection), thus $G/H$ is cyclic.


I just wanted to cite that more generally, if $G$ is produced by $n$ elements, climate every element group that $G$ is created by at many $n$ elements:

Let $G$ be generated by $\x_1,\ldots x_n\$, and let $N$ be a normal subgroup that $G$. Then every coset that $N$ in $G$ have the right to be expressed together a product of the cosets $Nx_1,\ldots, Nx_n$. For this reason the set $\Nx_1,\ldots,Nx_n\$ generates $G/N$, and this set contains at many $n$ elements.

(Note the the cosets $Nx_i$ will certainly not every be distinctive if $N$ is non-trivial, yet it"s well to create the set this way, simply as $\x^2 \mid x\in \thedailysplash.tvbbR\$ is a perfectly valid summary of the set of non-negative actual numbers.)

The an outcome about cyclic groups is then just the special instance $n=1$ the this.


Thanks for contributing an answer to thedailysplash.tvematics stack Exchange!

Please be sure to answer the question. Administer details and also share her research!

But avoid

Asking for help, clarification, or responding to various other answers.Making statements based upon opinion; ago them increase with references or an individual experience.

Use thedailysplash.tvJax to layout equations. Thedailysplash.tvJax reference.

See more: Tile Background Tumblr Pastel, Pink Tile Wallpaper/Background

To learn more, see our tips on writing great answers.

write-up Your answer Discard

By clicking “Post your Answer”, girlfriend agree to our terms of service, privacy policy and cookie plan

Not the price you're looking for? Browse various other questions tagged group-theory or questioning your own question.

site architecture / logo design © 2021 stack Exchange Inc; user contributions license is granted under cc by-sa. Rev2021.9.10.40187

your privacy

By clicking “Accept all cookies”, girlfriend agree ridge Exchange have the right to store cookie on your an equipment and disclose details in accordance through our Cookie Policy.