Mathematics
The Fundamental Theorem of Finite Abelian Groups
Quick fact
There are exactly 49 finite abelian groups of order 900, and the theorem provides a systematic way to count them—this count is only possible because every such group decomposes uniquely into a direct product of cyclic groups of prime-power order.
Why this is interesting
Have you ever wondered if two different-looking groups could secretly be the same? This theorem tells us exactly how to tell and lists all possible finite abelian groups.