The induced semi-direct product of a morphism.

Well there's this exercise:

Let $$\mathbb{K}^\times$$ be a commutative field.

Let $$\phi: \mathbb{K}^\times \times SL_n (\mathbb{K}) \rightarrow GL_n(\mathbb{K})$$ be an isomorphism (that I've proven), such that $$\phi(z,M)=\begin{vmatrix} z&0\\ 0&I_{n-1}\\ \end{vmatrix} M$$

What would be the induced function for the semi-direct product of $$\mathbb{K}^\times \rtimes_{f} SL_n (\mathbb{K})$$ ? (I don't understand what they mean by that, in french: "loi de produit semi-direct induite sur $$\mathbb{K}^\times \times SL_n (\mathbb{K})$$ par $$\phi$$")

Just for illustration:

Later on the exercise they ask me to show (using what we've shown) that $$S_n \cong A_n \rtimes_f P$$ where $$|P|=2$$ and $$P.

• As a set, the external semi-direct product is a Cartesian product and so isn't the induced function just itself? Nov 10, 2019 at 2:32
• Well I thought it should be something that defines the product: $(k_1,M_1) (k_2,M_2)=(k_1 \Phi_{M_1} (k_2),M_1M_2)$ Therefore: $\Phi: SL_n(\mathbb{K}) \rightarrow Aut(\mathbb{K})$ such that $\Phi(M_1) = \Phi_{M_1}$, so I would be searching this $\Phi_{M_1}$ that is induced by the original function defined before Nov 10, 2019 at 11:00
• Oh, I misunderstood what you meant by induced function. Here's a link to an answer to your question: math.stackexchange.com/a/670128/55622 Nov 10, 2019 at 22:39
• By the way, I think your map should be $\Phi : {\Bbb K}^\times \rightarrow Aut(SL_n({\Bbb K}))$. Nov 10, 2019 at 22:43