GATE 2025 MA – Question 64

Algebra · Groups · 2 marks · Numerical answer

In a group G with identity e, g≠e, g²=e, h≠e, h²≠e and ghg⁻¹=h². Find the least positive n with hⁿ=e (integer).

Show answer and explanation

Correct answer: 3 to 3

Explanation

Conjugate twice by g. Since g²=e, h=g(h²)g⁻¹=(ghg⁻¹)²=(h²)²=h⁴. Thus h³=e. As h≠e, its order is exactly **3**; the h²≠e condition is consistent.

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