Fundamental Theorem of Finite Abelian Groups

Abstract Algebra · Axiom Academy

LESSON Fundamental Theorem of Finite Abelian Groups Every finite abelian group can be uniquely decomposed into a direct product of cyclic groups of prime power order. This means any finite abelian group G can be written as: where each p_i is a prime and each . The group is the cyclic group of order p^k . 2. Understanding Cyclic Groups Before we can appreciate the theorem, let's recall what cyclic groups are: Why cyclic groups matter: They are the simplest abelian groups— completely determined by their order. They're the "atoms" from which all finite abelian groups are built. Let's see the theorem in action with a concrete example: . 4. Two Representations: Invariant Factors vs Elementary Divisors There are two ways to write the decomposition of a finite abelian group: Converting between them: For the same group of order 12: Elementary divisors: (prime powers: 2^1, 2^1, 3^1 ) Invariant factors: (where 2 | 6 ) The fundamental theorem lets us count how many distinct abelian groups exist of a given order! P(3) = 3 (partitions: 3 , 2+1 , 1+1+1 ) P(2) = 2 (partitions: 2 , 1+1 ) Total: distinct abelian groups of order 72

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