GATE 2024 MA – Question 27
How many nonisomorphic finite groups have exactly three conjugacy classes? (integer)
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Correct answer: 2 to 2
Explanation
The two groups are C₃ (three singleton classes) and S₃ (identity, three transpositions, two 3-cycles). To see there are no others, write class sizes 1,a,b with a≤b. Their centralizer orders m≥k≥2 satisfy 1/|G|+1/m+1/k=1. The integer possibilities force |G|=3,4 or 6; order-four groups are abelian and have four classes. Order three gives C₃, and only the nonabelian order-six group S₃ has three classes. Thus **2**.