Classification of Finite Abelian Groups

Every finite abelian group is a direct product of cyclic prime-power groups, uniquely up to isomorphism.

Classification of Finite Abelian Groups

Classification of Finite Abelian Groups: Let $G$ be a finite abelian group . Then $G$ is isomorphic to a finite direct product of cyclic groups of prime-power order:

G \cong \prod_{i=1}^t \prod_{j=1}^{r_i} C_{p_i^{e_{ij}}},

where $p_1,\dots,p_t$ are primes, each $e_{ij}\ge 1$ , and $C_{p^e}$ denotes a cyclic group of order $p^e$ (a finite p-group ).

Equivalently (the invariant factor form), there exist integers $1<n_1\mid n_2\mid \cdots \mid n_r$ such that

G \cong C_{n_1}\times C_{n_2}\times \cdots \times C_{n_r}.

In either form, the invariants (prime powers in the first form, or the chain $n_1\mid\cdots\mid n_r$ in the second form) are uniquely determined by $G$ up to reordering of isomorphic factors.

This is the finite-group specialization of the fundamental theorem of finitely generated abelian groups (since every finite group is finitely generated).

Examples:

Order $8=2^3$ $8 = 2^{3}$ : the abelian groups of order $8$ $8$ are, up to isomorphism,
- $C_8$ ,
- $C_4\times C_2$ ,
- $C_2\times C_2\times C_2$ .
Order $12=2^2\cdot 3$ $12 = 2^{2} \cdot 3$ : the abelian groups of order $12$ $12$ are, up to isomorphism,
- $C_{12}\cong C_4\times C_3$ ,
- $C_2\times C_2\times C_3 \cong C_2\times C_6$ .