Question

Determine by checking the group axiom (Associativity) whether this set is a group:

(C, o), where x o y = x + y - 2i

C = set of all imaginary numbers

Answer 1

Let x, y, z ∈ C, the set of complex numbers.

By definition of ○,

(x ○ y) ○ z = (x + y - 2i) ○ z

… = (x + y - 2i) + z - 2i

… = x + y + z - 4i

and

x ○ (y ○ z) = x ○ (y + z - 2i)

… = x + (y + z - 2i) - 2i

… = x + y + z - 4i

so (x ○ y) ○ z = x ○ (y ○ z), which means ○ is indeed associative under C.