# Thread: Isomorphism between D4 and Z2 x Z4

1. ## Isomorphism between D4 and Z2 x Z4

Show that $D_4$ and $\mathbb{Z}_2$ x $\mathbb{Z}_4$ are not isomporphic

Be grateful for help with the approach to this problem.

For those who need an update or reaquaintance with $D_4$ I have attached the relevant pages of Fraleigh: A First Course in Abstract Algebra.

Peter

2. ## Re: Isomorphism between D4 and Z2 x Z4

Originally Posted by Bernhard

Show that $D_4$ and $\mathbb{Z}_2$ x $\mathbb{Z}_4$ are not isomporphic

Be grateful for help with the approach to this problem.

For those who need an update or reaquaintance with $D_4$ I have attached the relevant pages of Fraleigh: A First Course in Abstract Algebra.

Peter
It is known - or easy to prove - that homomorphic image of an abelian group is an abelian group, too.

Notice that $D_4$ is not abelian while $\mathbb{Z}_2 \times\mathbb{Z}_4$ is abelian.

3. ## Re: Isomorphism between D4 and Z2 x Z4

Originally Posted by zoek
It is known - or easy to prove - that homomorphic image of an abelian group is an abelian group, too.

Notice that $D_4$ is not abelian while $\mathbb{Z}_2 \times\mathbb{Z}_4$ is abelian.
Another approach would be to note that that isomorphisms are bijections. Therefore, they preserve order. So, $\mathbb{Z}_2\times \mathbb{Z}_4$ has order 8, while the Klein 4-group has order 4. So...

4. ## Re: Isomorphism between D4 and Z2 x Z4

Thanks for the post

But Swlabr. this post concerns $D_4$ which has order 8 ... not the Klein 4-group (order 4) ...

Peter

5. ## Re: Isomorphism between D4 and Z2 x Z4

Originally Posted by Bernhard
Thanks for the post

But Swlabr. this post concerns $D_4$ which has order 8 ... not the Klein 4-group (order 4) ...

Peter
Ah, sorry, there are two conventions - one can use either $D_{n}$ or $D_{2n}$ for the Dihedral group of order $2n$. I should have realised from context which one you were using...

6. ## Re: Isomorphism between D4 and Z2 x Z4

oh ... understand!

Peter

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### order of z2xz4

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