Why are these groups isomorphic?

Question

I have this group of permutations:

And I have this group of complex numbers:

These groups are isomorphic to each other, but it seems I do not understand why. I was looking for similarities in structure and I would have thought because they do not share the same structure, that they were not isomorphic. Like take the diagonal of $g$s in the group of permutations. In the group of complex numbers this same diagonal is made of different elements, $i$ and $1$. It seems

I do not fully understand isomorphism, because I was looking for the same structure. Could someone explain why these groups are isomorphic?

Answer 1

You can see it by constructing an explicit bijective homomorphism.

$e\leftrightarrow 1$, $g\leftrightarrow -1$, and then $f\leftrightarrow i$ or $-i$ and $h$ is whatever remaining that isn't $f$.

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Alternatively, you can show that both groups are cyclic of order four.

$i\mapsto i^2=-1\mapsto i^3=-i\mapsto i^4=1\mapsto i^5=i$

$f\mapsto f^2=g\mapsto f^3=h\mapsto f^4=e\mapsto f^5=f$

Answer 2

Hint: For the second group, use the order $1,i,-1,-i$.

Answer 3

They are both isomorphic to they cyclic group of order 4, with $f$ as a generator of the first group, and $i$ a generator of the group of complex numbers.