Prove that exactly one of the non-abelian groups of order 8 is isomorphic to a semidirect product C4 x C₂, explicitly identifying which one it is.

Let G = R – {-1}. Define the binary operation * on G by x * y = x +y+xy. a) Show that (G, *) is an abelian group. b) Let H = R – {0}. Show that (G, *) and (H, ·) are isomorphic where · represents the usual product on R.

(4) Prove that Z x Z/((2,2)) is an infinite group but is not an infinite cyclic group.

If R2 is the plane considered as an (additive) abelian group, show that any line L through the L in R2. Provide a origin is a subgroup. Now, if BE R2, then 8+ L represents the left coset geometric description of this coset and explain how you arrived at this description.