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I am working on a problem and at some point i wanted to check if there are any homomorphisms (non-trivial) from $Z_{7}$ to $\operatorname{Aut}( Z_{4} \times Z_{2})$ I was hoping there won't be any but i got the automorphisms explicitly ( 8 of them) and I found 5 such homomorphisms. Is that right? Or do i have a mistake somewhere?


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up vote 4 down vote accepted

$\mathbb Z_7$ is a cyclic group of order $7$. Its image must be trivial or a cyclic group of order $7$. If you found $8$ automorphisms, the order of all elements in $\operatorname{Aut}( \mathbb Z_{4} \times\mathbb Z_{2})$ must divide $8$, so it can't be $7$. Thus the image can only be trivial.

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