Let S = R\{-1} and define a binary operation on S by a * b = a +b+ ab. Prove that S is an abelian group. Note: you must prove that this structure has all of the properties of a group (associativity, identity, inverses), and also that the operation is commutative.

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.

is it a binary option? is it associative? identity element= inverse elements=

Prove that a group is abelian if and only if (ab)-) ab)1= a6 va,bEG Attach File Browse Local Files

Verify that (ℤ, ⨀) is an infinite group, where ℤ is the set of integers and the binary operator ⨀ is defined asa⨀ b = a2 + b

Give an example of a finite group that is not abelian. (will leave a like)