semidirect product কী?

Example 1.27

(a) The dihedral group $D_n$ ($n \geq 3$) of the $2n$ symmetries of a regular $n$-gon: the rotations form a normal subgroup of index $2$, any reflection generates a complement, and conjugating a rotation by a reflection inverts it: $D_n \cong \Z/n\Z \rtimes_\varphi \Z/2\Z$ with $\varphi(1) = (x \mapsto -x)$. (b) The affine group of a line, $\{x \mapsto ax + b : a \in K^\times,\, b \in K\} \cong K \rtimes K^\times$: translations normal, homotheties a complement. (c) $S_n \cong A_n \rtimes \Z/2\Z$ (complement: any transposition). (d) The quaternion group $Q_8$ is not a semidirect product of proper subgroups: every nontrivial subgroup contains $-1$ (Problem 1.1), so no two proper subgroups intersect trivially.