LetS=R{−1} and define a binary operationon S  by a∗b=a+b+ab. Prove that (S, ∗) is an abelian group.

Asap

Show that (4Z) (6Z) is a commutative group.

12. Let Z₂ = {(a₁, a2, … ..‚ an) : a¡ € Z2}. Define a binary operation on Zby (a₁, a2,..., an) + (b₁,b₂, ..., bn) = (a₁ + b₁, a2 + b2,..., an+bn). Prove that Z2 is a group under this operation. This group is important in algebraic coding theory.

Please don't provide handwritten solution...

Determine the Galois group of x2 + 9 over R.

Let G1 and G2 be groups. Prove that G1 × G2 is abelian if and only if G¡ and G2 are both abelian.

Use the fundamental theorem of Abelian groups to express Z20 as an external direct product of cyclic groups of prime power order.