Solution for Section G.1 Question 6

6. Check the group axioms:

• closure under multiplication modulo p: if you multiply any pair of elements in Z_p-{0}, is the result also an element of Z_p-{0}?
  Z_p-{0} with multiplication modulo p is closed. When two elements [a] and [b] in Z_p-{0} are multiplied, the result is reduced modulo p so the answer will lie in the range [0] to [p-1]. We need to show that the answer will not be [0] since [0] is not in Z_p-{0}. Note that [0] = [rp] for any integer r. Since p is prime, the only way to have a × b = r × p, would be if p divided one of a or b. We cannot have this, so [a] × [b] will never give [0].
  
• associativity: does (a × b) × c = a × (b × c) for a, b, c Z_p-{0}?
  Multiplicaion is associative, so multiplication modulo p is associative.
  
• identity: do we have e Z_p-{0} such that e × a = a = a × e for all a Z_p-{0}?
  [1] is the identity element in Z_p-{0}.
  
• inverse: for all a Z_p-{0}, do we have an a^-1 such that a × a^-1 = e = a^-1 × a?
  Every element in Z_p-{0} has a multiplicative inverse. This was shown in Question 4.

Back to Section G.1