Going over an exercise in Dummit and Foote (5.5 #12) and got to a point where I have to show the 3 non-abelian groups of order 20. I was thinking about it more and it seems like the group $C_5 \times C_4$ is isomorphic to all the semi-direct products between $C_5$ and $C_4$. I think this would be because all the homomorphisms $\phi : C_4 \rightarrow Aut(C_5)$ have images which are conjugates of each other so there's really just the one semidirect product and the trivial semidirect product (the direct product). But the non-trivial semidirect products are still isomorphic to a direct product with an order 5 cyclic group in one spot and an order 4 cyclic group in the other. Is this true? Is there some intuition or angle that I'm missing here?
2026-03-25 18:49:06.1774464546
Are the semidirect products of two finite cyclic groups all isomorphic?
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