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Let G and K be two groups and $\phi_1$ and $\phi_2: G \rightarrow Aut(K)$ be homomorphism.

Q1: If $\phi_1$ not trivial homomorphism, can When can semidirect product of G and K using $\phi_1$ isomorphic to direct product of G and K?

Q2: Will semidirect product of G and K using $\phi_1$ and $\phi_2$ be non- isomorphic?

Q3: What will be answer to Q1 and Q2 if both groups G and K are finite cyclic groups or cyclic groups?

Q4: If answer to Q2 is no, when can semidirect product of G and K using $\phi_1$ and $\phi_2$ can be isomorphic?

And let me mention these are not homework problems.

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  • $\begingroup$ "can When can"? And if you meant to ask whether $\;K\rtimes_{\phi_1} G\cong G\times K\;$ then this is so only if $\;\phi_1\;$ is trivial. There are also conditions when two semidirect products are isomorphic. For example, if $\;\phi_1,\phi_2\;$ are conjugated by an element in Aut$\,(K)\;$ , et.c $\endgroup$ – DonAntonio Jun 16 '16 at 12:31
  • $\begingroup$ @Joanpemo It is possible to have $K \rtimes_{\phi} G \cong K \times G$ when $\phi$ is nontrivial but ${\rm im}(\phi) \le {\rm Inn}(K)$. $\endgroup$ – Derek Holt Jun 16 '16 at 12:44
  • $\begingroup$ @DerekHolt Thank you, didn't know that. Could you please give some example? $\endgroup$ – DonAntonio Jun 16 '16 at 12:46
  • $\begingroup$ Let $K=G$ be any nonabelian group and define $\phi(g)$ to be conjugation by $g$. Then $K \rtimes_\phi G \cong K \times G$. See math.stackexchange.com/questions/201710 $\endgroup$ – Derek Holt Jun 16 '16 at 13:10
  • $\begingroup$ haha atleast counterexample you gave for Q2 but what happens in case of abelian groups I should also ask. I have only asked for cyclic groups in Q 3 @DerekHolt $\endgroup$ – Sushil Jun 16 '16 at 15:02

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