Show the quaternion group as a semidirect product

Question

Question:

Semidirect product

If H and K are subgroup of G, with K a normal subgroup of G, K intersect H=1 and KH=G, then G is called a semidirect product (or split extention) of K by H.

If sigma=(12) in S_n, n>=2, show that s_n is a semidirect product of A_n by

Show that the dihedral group D_n=n=b²=1,b^-1ab=a^-1> is a semidirect product of A= by B=

Show that the quaternion group Q₂ cannot be expressed as a semidirect product of two non-trivial subgroups.