GATE 2020 MA – Question 62
G is a noncyclic group of order 57. Find its number of elements of order 3.
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Correct answer: 38 to 38
Explanation
Sylow’s theorem gives n₁₉ dividing 3 and congruent to 1 modulo 19, hence n₁₉ = 1. Also n₃ divides 19 and is 1 modulo 3, so n₃ = 1 or 19. If both Sylow subgroups were normal, their coprime-order direct product would be cyclic of order 57, contrary to the premise. Thus n₃ = 19. Distinct subgroups of order 3 intersect only at the identity and each contributes two order-three elements, giving 19×2 = 38.