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This is the question with its answer:enter image description here

It is not clear for me why we should show the first statement in the following picture:enter image description here

Could anyone explain this for me please?

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    $\begingroup$ Actually the offered solution is wrong when it says II (or I) follows from III -- that would be the case if III claimed that all the maps $z\mapsto z^k$ are homomorphisms, but actually it only claims that all homomorphisms are among those maps. $\endgroup$ – hmakholm left over Monica Jul 1 '17 at 14:33
  • $\begingroup$ @HenningMakholm You're right, but actually in this case $z\mapsto z^k$ is homomorphism $\forall k$. Of course, III doesn't say it. $\endgroup$ – user261263 Jul 1 '17 at 15:16
  • $\begingroup$ @EugenCovaci: True enough, of course; it is just not what III in the problem states. $\endgroup$ – hmakholm left over Monica Jul 1 '17 at 15:20
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Since $i$ generates the group then $z=i^m$ for some $m$. Now, if $i$ maps to $i^k$ then $z$ maps to $i^{mk}=z^k$ following the homomorphism properties.

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