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GATE CSE 2024 Session 2 Q63: Self-inverse elements of Z2 × Z3 × Z4

Question

Let ZnZ_n be the group of integers {0,1,2,…,n−1}\{0, 1, 2, \ldots, n - 1\} with addition modulo nn as the group operation. The number of elements in the group Z2×Z3×Z4Z_2 \times Z_3 \times Z_4 that are their own inverses is ________

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Numerical answer. No negative marking.
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Answer: 4 (official GATE 2024 final answer key)

Concept used: In a direct product, an element is its own inverse exactly when each component is.

First step: Count solutions of 2g=02g = 0 in each factor: Z2Z_2 has 2, Z3Z_3 has 1, Z4Z_4 has 2.

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Source: official GATE 2024 Session 2 CS question paper and final answer key, organised by IISc Bangalore. Question paper · Answer key

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