FORM GR1268 Question 8

Question:
Which of the following is NOT a group?
(A) The integers under addition
(B) The nonzero integers under multiplication
(C) The nonzero real numbers under multiplication
(D) The complex numbers under addition
(E) The nonzero complex numbers under multiplication

Answer:
(B)

Answer Key:
Since all the answer choices deal with numbers, associativity holds.

For (A):
Identity is \(0\).
Inverse of \(x\in\mathbb Z\) is \(-x\in\mathbb Z\).
Integer plus integer is integer (closure).

For (B):
Identity is \(1\).
Inverse of \(x\in\mathbb Z\setminus\{0\}\) is \(\frac{1}{x}\notin\mathbb Z\setminus\{0\}\).
Integer times integer is integer (closure).

For (C):
Identity is \(1\).
Inverse of \(x\in\mathbb R\setminus\{0\}\) is \(\frac{1}{x}\in\mathbb R\setminus\{0\}\).
Non-zero real number times non-zero real number is non-zero real number (closure).

For (D):
Identity is \(0\).
Inverse of \(x+iy\in\mathbb C\) is \(-x+i(-y)\in\mathbb C\).
Complex number plus complex number is complex number (closure).

For (E):
Identity is \(1\).
Inverse of \(x+iy\in\mathbb C\setminus\{0\}\) is
\[\displaystyle\frac{1}{x+iy}=\frac{x-iy}{x^2+y^2}=\frac{x}{x^2+y^2}+i\left(-\frac{y}{x^2+y^2}\right)\in\mathbb C\setminus\{0\}.\]
Non-zero complex number times non-zero complex number is non-zero complex number (closure).