Solution for Section G.1 Question 7

7. Check the group axioms:

• closure under *: if a and b are in Z, is a * b also in Z?
  For any a,b in Z, a*b = a+b+2 is also in Z. Hence Z is closed under *.
  
• associativity: does (a * b) * c = a * (b * c) for a, b, c Z?
  For any a,b,c in Z,
  we have (a * b) * c = (a + b + 2) * c = (a + b + 2) + c + 2 = a + b + c + 4
  and a * (b * c) = a * (b + c + 2) = a + (b + c + 2) + 2 = a + b + c + 4.
  Hence * is associative.
• identity: do we have e Z such that e * a = a = a * e for all a Z?
  For any a in Z, a * (-2) = a + (-2) + 2 = a, and (-2) * a = (-2) + a + 2 = a. Thus -2 is the identity element.
• inverse: for all a Z, do we have an a^-1 such that a * a^-1 = e = a^-1 * a?
  For any a in Z, a * (-a-4) = a + (-a-4) + 2 = -2, and (-a-4) * a = (-a-4) + a + 2 = -2. Thus -a-4 is the inverse of element a.
• commutativity: does a * b = b * a for a, b Z?
  For any a,b in Z,
  we have a * b = a + b + 2
  and b * a = b + a + 2.
  Since a + b + 2 = b + a + 2, the operation * is commutative.

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