GATE CS 2024 Set 2 Q.63: Monoids and Groups

Let $Z_n$ be the group of integers $\{0, 1, 2, \dots, n - 1\}$ 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 ________

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Official answer and solution

Official answer: 4
From: Official GATE 2024 answer key

Solution

Count the self-inverse elements ( $2x \equiv 0 \pmod n$ ) in each factor group $Z_2, Z_3, Z_4$ and multiply the counts.

An element $(a, b, c) \in Z_2 \times Z_3 \times Z_4$ is its own inverse if and only if $(a, b, c) + (a, b, c) = (0, 0, 0)$ , which requires each component to be its own inverse in its respective factor group:

In $Z_2$ ( $2a \equiv 0 \pmod 2$ ): $a \in \{0, 1\}$ (2 elements).
In $Z_3$ ( $2b \equiv 0 \pmod 3$ ): $b \in \{0\}$ (1 element).
In $Z_4$ ( $2c \equiv 0 \pmod 4$ ): $c \in \{0, 2\}$ (2 elements).

By the multiplication principle, the total number of self-inverse elements in $Z_2 \times Z_3 \times Z_4$ is:

2 \times 1 \times 2 = 4

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2024 Computer Science and Information Technology - Set 2 (CS2)
Question: 63 of 65
Section: Computer Science and Information Technology

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