GATE 2024 CS2 Q.63: solution and answer

GATE 2024 · CS2Q.63 · 2 marks · NAT

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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Answer and solution

ANSWER

There are 4 elements of $Z_{2} \times Z_{3} \times Z_{4}$ that are their own inverses.

Official answer: 4Source: Official answer key, GATE 2024 CS2 (IISc Bengaluru)

An element $g$ is its own inverse exactly when $g + g = e$ , that is, $2 g = e$ .
The operation in a direct product is componentwise, so $(a, b, c)$ satisfies $2 (a, b, c) = (0, 0, 0)$ iff $2 a \equiv 0 (mod 2)$ , $2 b \equiv 0 (mod 3)$ and $2 c \equiv 0 (mod 4)$ . The three conditions are independent, so the choices can be counted separately.
In $Z_{2}$ : $2 a \equiv 0 (mod 2)$ holds for every $a$ , so there are 2 choices ( $0, 1$ ).
In $Z_{3}$ : $2 b \equiv 0 (mod 3)$ forces $b = 0$ because $g cd (2, 3) = 1$ , so there is 1 choice.
In $Z_{4}$ : $2 c \equiv 0 (mod 4)$ gives $c \in {0, 2}$ , so there are 2 choices.
Multiply the independent choices: $2 \times 1 \times 2 = 4$ .
The four elements are $(0, 0, 0), (1, 0, 0), (0, 0, 2), (1, 0, 2)$ .

KEY IDEA

In a direct product of groups the operation acts on each coordinate separately, so the elements with $2 g = e$ can be counted coordinate by coordinate. In $Z_{n}$ the solutions of $2 x \equiv 0 (mod n)$ number $g cd (2, n)$ , which gives $2 \cdot 1 \cdot 2 = 4$ .

WATCH OUT FOR

Counting only the identity, or taking the self-inverse elements of $Z_{3}$ to be ${0, 1, 2}$ or the ones of $Z_{4}$ to be just ${0}$ . Another error is adding the counts (2+1+2=5) instead of multiplying them.

From the GATE 2024 CS2 paper. Every question of this paper · Download the full paper · All 7 Groups questions, with notes

GATE 2024 CS2 Q.63: solution and answer

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