# Tagged: semidirect product

## Problem 465

Prove that the symmetric group $S_n$, $n\geq 3$ is a semi-direct product of the alternating group $A_n$ and the subgroup $\langle(1,2) \rangle$ generated by the element $(1,2)$.

## Problem 113

Let $A$, $B$ be groups. Let $\phi:B \to \Aut(A)$ be a group homomorphism.
The semidirect product $A \rtimes_{\phi} B$ with respect to $\phi$ is a group whose underlying set is $A \times B$ with group operation
$(a_1, b_1)\cdot (a_2, b_2)=(a_1\phi(b_1)(a_2), b_1b_2),$ where $a_i \in A, b_i \in B$ for $i=1, 2$.

Let $f: A \to A’$ and $g:B \to B’$ be group isomorphisms. Define $\phi’: B’\to \Aut(A’)$ by sending $b’ \in B’$ to $f\circ \phi(g^{-1}(b’))\circ f^{-1}$.

$\require{AMScd} \begin{CD} B @>{\phi}>> \Aut(A)\\ @A{g^{-1}}AA @VV{\sigma_f}V \\ B’ @>{\phi’}>> \Aut(A’) \end{CD}$ Here $\sigma_f:\Aut(A) \to \Aut(A’)$ is defined by $\alpha \in \Aut(A) \mapsto f\alpha f^{-1}\in \Aut(A’)$.
Then show that
$A \rtimes_{\phi} B \cong A’ \rtimes_{\phi’} B’.$