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Improved the title and clean the body
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edit approved Apr 22, 2021 at 20:45
Mike Pierce
edit approved Apr 22, 2021 at 20:45
Mike Pierce
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Semidirect Products Do isomorphic semi-direct products correspond to conjugate automorphisms?

Let H, N$H$ and $N$ be two groups with H$H$ cyclic. Let $f,g:H \rightarrow Aut(N)$$f,g:H \rightarrow \mathrm{Aut}(N)$ be homomorphisms such that $N\rtimes _f H \cong N\rtimes _g H$. Then does that mean $f(H)$ and $g(H)$ are conjugate in $Aut(N)$$\mathrm{Aut}(N)$?

Semidirect Products

Let H, N be two groups with H cyclic. Let $f,g:H \rightarrow Aut(N)$ be homomorphisms such that $N\rtimes _f H \cong N\rtimes _g H$. Then does $f(H)$ and $g(H)$ are conjugate in $Aut(N)$?

Do isomorphic semi-direct products correspond to conjugate automorphisms?

Let $H$ and $N$ be two groups with $H$ cyclic. Let $f,g:H \rightarrow \mathrm{Aut}(N)$ be homomorphisms such that $N\rtimes _f H \cong N\rtimes _g H$. Then does that mean $f(H)$ and $g(H)$ are conjugate in $\mathrm{Aut}(N)$?

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Soluble
Soluble
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Semidirect Products

Let H, N be two groups with H cyclic. Let $f,g:H \rightarrow Aut(N)$ be homomorphisms such that $N\rtimes _f H \cong N\rtimes _g H$. Then does $f(H)$ and $g(H)$ are conjugate in $Aut(N)$?