Question:
The number of non-isomorphic finite groups with exactly 3 conjugacy classes is equal to (answer in integer):
The number of non-isomorphic finite groups with exactly 3 conjugacy classes is equal to (answer in integer):
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For group theory problems, analyze the number of conjugacy classes and the structure of the group.
Updated On: Feb 1, 2025
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Solution and Explanation
Step 1: Analyzing group order.
Finite groups with exactly 3 conjugacy classes must have order \( p^2 \) for some prime \( p \).
Step 2: Counting non-isomorphic groups.
For order \( p^2 \), there are exactly two non-isomorphic groups:
1. The cyclic group \( {Z}_{p^2} \).
2. The direct product \( {Z}_p \times {Z}_p \).
Step 3: Conclusion.
The number of non-isomorphic groups is \( {2} \).
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