isomorphic group

**grace412** · May 5th 2011, 07:21 AM

Is 1Z/14Z isomorphic to 2Z/28Z? How do I solve or prove this?

**Swlabr** · May 5th 2011, 08:21 AM

Originally Posted by grace412

Is 1Z/14Z isomorphic to 2Z/28Z? How do I solve or prove this?

Yes. Think about the order of your groups. If x is the generator of Z, then in Z/14Z it(s image) has order 14 (obviously), so what is the order of (the image of) 2x in 2Z/28Z? Both groups are cyclic, so...

(Another way you can think about this is index: What is the index of 28Z in 2Z? and 14Z in Z? As images of cyclic groups are cyclic then...)

**abhishekkgp** · May 5th 2011, 08:32 AM

Originally Posted by grace412

Is 1Z/14Z isomorphic to 2Z/28Z? How do I solve or prove this?

define a map
$\phi:\mathbb{Z}/14\mathbb{Z} \rightarrow 2\mathbb{Z}/ 28\mathbb{Z}$ defined as:
$k+14\mathbb{Z} \mapsto 2k+28\mathbb{Z}$

show now that $\phi$ is homomorphism, injection and surjection.

**Deveno** · May 5th 2011, 10:51 PM

if you follow the advice of the post above, be sure to also show that $\phi$ is well-defined.

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