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Is $ G = \frac{\mathbb{Z}_{2} \times \mathbb{Z}_4 \times \mathbb{Z}_{8}}{<(1,2,4)>}$ isomorphic to $\mathbb{Z}_8 \times \mathbb{Z}_4$?

I thought I might try reducing $\mathbb{Z}_{2} \times \mathbb{Z}_4 \times \mathbb{Z}_{8}$ into more decomposition and then try reducing my options from there, but its already in its most reduced form. How else though can we try this?

Since the order of our factor group is $32$, I wrote down all of decomposition of $\mathbb{Z}_{32}$

Ozera
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The quickest way to show isomorphism is to find a surjective group morphism $f:\mathbb Z_2\times\mathbb Z_4\times\mathbb Z_8\to\mathbb Z_8\times\mathbb Z_4$ such that $\ker f=\langle(1,2,4)\rangle$.

A little search shows that $f(x,y,z)=(z+4x,y+2x)$ seems a good candidate, where $\mathbb Z_2$ is identified with $\{0,1\}$, i.e. $tx=0$ if $x=0$ and $t$ if $x=1$.

Try to verify that it is a morphism, and that the kernel is what you want.

Denis
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  • I don't think I've ever done an exercise relating to this that way. I thought I would check something relating to orders...but not sure – Ozera May 06 '14 at 14:52
  • having same order is necessary but not sufficient to be isomorphic – Denis May 06 '14 at 15:03
  • Can you do an argument like this one: http://math.stackexchange.com/q/390488/80327 ? I think I would understand it better – Ozera May 06 '14 at 15:09
  • wow it's a lot more complicated. in my answer you just need to know that if $f: G_1\to G_2$ is a surjective morphism, then $\frac{G_1}{\ker f}$ is isomorphic to $G_2$, which is a very classical theorem that if you didn't learn it already you will soon. – Denis May 06 '14 at 15:13
  • To be clear, I speaking of the solution that the OP gave. – Ozera May 06 '14 at 15:18
  • @Denis, perhaps one of the doubts Ozera has here is why did you choose $;\Bbb Z_8\times \Bbb Z_4;$ to define your homomorpishm to? Why not $;\Bbb Z_2\times\Bbb Z_4\times\Bbb Z_4;$ , or $;\Bbb Z_2\times\Bbb Z_2\times\Bbb Z_8;$ , say? – DonAntonio May 06 '14 at 15:55
  • @DonAntonio because that is the question of the OP... – Denis May 06 '14 at 16:03