No Simple Groups of Order 66 or 144

Explore an advanced group theory problem using Sylow's Theorems to prove the non-existence of simple groups with orders 66 and 144. Delve into the application of these powerful theorems to analyze group structures and demonstrate why no simple groups can exist at these specific orders. Gain insights into the fundamental principles of abstract algebra and enhance your understanding of group theory through this rigorous mathematical exploration.

No simple groups of order 66 or 144.