Let \(Z_n\) be the group of integers \(\{0, 1, 2, \ldots, n - 1\}\) with addition modulo \(n\) as the group operation. The number of elements in the group \(Z_2 \times Z_3 \times Z_4\) that are their own inverses is ________
Answer
4
Explanation
The product group works coordinate by coordinate, so \((x, y, z)\) is its own inverse exactly when \(2x \equiv 0 \pmod 2\), \(2y \equiv 0 \pmod 3\) and \(2z \equiv 0 \pmod 4\).
- In \(Z_2\): both 0 and 1 work (2 choices).
- In \(Z_3\): only 0 works, since \(2 \cdot 1 = 2\) and \(2 \cdot 2 = 4 \equiv 1\) (1 choice).
- In \(Z_4\): 0 and 2 work (2 choices).
So there are \(2 \times 1 \times 2 = 4\) such elements: \((0,0,0), (0,0,2), (1,0,0), (1,0,2)\). The identity \((0,0,0)\) counts, since it is its own inverse.
Answer: 4