What is a semi-direct product of two groups? What is an example of a semi-direct product that is not a direct product?
From what I read in the textbook, a semi-direct product N $\rtimes_\alpha$ A, contains A and N as subgroups with N normal. N $\cap$ A = {$e$},
N$\times$A= N$\rtimes_\alpha$A and there is a commutation relation $a_n = \alpha_a(n)a$ where $a\in A$, $n\in N$.
$(n,a)(n',a')=(n\alpha_a(n'),aa')$.
Is this correct? Also, that a direct product is just a special case of a semi-direct product. How would I find an example of a semi-direct product that is not a direct product ?
One simple example is the dihedral group.
As a special case, the symmetric group $S_3$ is isomorphic to $D_3$.