GATE CS 2024 Set 2 — Question 63

NAT+2 / -0MediumGroups, Rings & FieldsSets & CombinatoricsEngineering Mathematics

Engineering Mathematics → Sets & Combinatorics → Groups, Rings & Fields

Last updated 20 August 2026

Question

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 ________

4 to 4

An element

(a, b, c)

in the group

G = Z_2 \times Z_3 \times Z_4

is its own inverse if

(a, b, c) + (a, b, c) = (0, 0, 0)

, where

(0, 0, 0)

is the identity element of the group. This condition is equivalent to solving the following congruences independently for each component:

1.In $Z_2$ :

2a \equiv 0 \pmod 2

. This is satisfied by

a = 0

and

a = 1

. Thus, there are 2 such elements in

Z_2

2.In $Z_3$ :

2b \equiv 0 \pmod 3

. Since

\gcd(2, 3) = 1

, the only solution is

b = 0

. Thus, there is 1 such element in

Z_3

3.In $Z_4$ :

2c \equiv 0 \pmod 4

. This is satisfied by

c = 0

and

c = 2

. Thus, there are 2 such elements in

Z_4

According to the fundamental principle of counting, the total number of elements in the product group

Z_2 \times Z_3 \times Z_4

that are their own inverses is the product of the number of solutions in each component:

\text{Total elements} = 2 \times 1 \times 2 = 4

The specific elements are

(0, 0, 0), (0, 0, 2), (1, 0, 0), \text{ and } (1, 0, 2)