Fundamental Theorem of Finite Abelian Groups

Theorem

Every finite abelian group $G$ is isomorphic to a direct product of cyclic groups of prime-power order:

$$G\cong C_{p_1^{r_1}}\times C_{p_2^{r_2}}\times \cdots \times C_{p_k^{r_k}},$$

where the $p_i$ are (not necessarily distinct) primes. The decomposition is unique up to reordering.

What It Says

Every finite abelian group factors uniquely as a direct product of cyclic prime-power pieces. So the only finite abelian groups, up to isomorphism, are pinned down by the multi-set of prime-power orders.

Building Block: Products of Coprime Cyclic Groups

The factors above are written as prime powers, but you can recombine coprime ones using:

Proposition

Let $C_n,C_m$ be finite cyclic groups. Then

$$C_n\times C_m\cong C_{nm}⟺\gcd (m,n)=1.$$

When $\gcd (m,n)=1$, the subgroups $⟨{\rho }^m⟩\cong C_n$ and $⟨{\rho }^n⟩\cong C_m$ of $C_{nm}=⟨\rho ⟩$ have trivial intersection and product $C_{nm}$, so $C_{nm}$ is their Internal direct product. When $\gcd (m,n)=d>1$ no element of $C_n\times C_m$ reaches order $nm$ (every element is killed by $mn/d$), so the product is not cyclic.

Example

$C_{80}\cong C_{16}\times C_5$ and $C_7\times C_{25}\cong C_{175}$, but $C_2\times C_2\ncong C_4$.

This is why the “Warning!” in the notes matters: the same group can be written as $C_{5^4}\times C_{5^2}\times C_{3^2}\times C_3\times C_2$ (prime-power form) or recombined as $C_{11250}\times C_{75}$ - both correct, just different conventions.

Abelian $p$-Groups

The classification for a single prime is the heart of the proof:

Theorem

Every finite abelian group of order $p^n$ is isomorphic to $C_{p^{e_1}}\times \cdots \times C_{p^{e_t}}$ with $p^{e_1}\cdots p^{e_t}=p^n$, unique up to reordering.

So the abelian groups of order $p^n$ correspond exactly to the integer partitions of $n$ (the “shapes” $(e_1,\dots ,e_t)$). For example, the partitions of $3$ give the three groups of order $8$.

Counting Abelian Groups of a Given Order

Number of abelian groups of order $p^n$ = number of integer partitions of $n$.

| $|G|$ | Factorisation | Abelian groups (up to isomorphism) | |---|---|---| | $4$ | $2^2$ | $C_4$, $C_2\times C_2$ | | $8$ | $2^3$ | $C_8$, $C_4\times C_2$, $C_2\times C_2\times C_2$ | | $9$ | $3^2$ | $C_9$, $C_3\times C_3$ | | $12$ | $2^2\cdot 3$ | $C_4\times C_3\cong C_{12}$, $C_2\times C_2\times C_3\cong C_2\times C_6$ | | $16$ | $2^4$ | $C_{16}$, $C_8\times C_2$, $C_4\times C_4$, $C_4\times C_2\times C_2$, $C_2^4$ |

Proof Sketch

Two steps:

1. Sylow decomposition. An abelian $G$ has all subgroups normal, so its Sylow $p$-subgroups are normal and pairwise have trivial intersection. Hence

$$G\cong P_{p_1}\times P_{p_2}\times \cdots \times P_{p_k},$$

where $P_{p_i}$ is the Sylow $p_i$-subgroup. 2. Each $p$-group decomposes into cyclic factors. A finite abelian $p$-group factors as $C_{p^{r_1}}\times C_{p^{r_2}}\times \cdots$ (induction on order, using a maximal-cyclic-subgroup argument).

The uniqueness up to reordering follows from comparing prime-power-component counts in any two decompositions.

Equivalent Formulation: Invariant Factors

Equivalently, $G\cong C_{n_1}\times C_{n_2}\times \cdots \times C_{n_s}$ with $n_1\mid n_2\mid \cdots \mid n_s$. The $n_i$ are the invariant factors of $G$. The two formulations (prime-power decomposition vs invariant factors) are interchangeable via the Chinese Remainder Theorem.

Finitely Generated Version

The theorem extends from finite to finitely generated abelian groups by allowing free $\mathbb{Z}$-factors:

Fundamental Theorem of Finitely Generated Abelian Groups

Every nontrivial finitely generated abelian group $G$ is isomorphic to

$${\mathbb{Z}}^{e_0}\times C_{p_1^{e_1}}\times \cdots \times C_{p_n^{e_n}},$$

unique up to reordering. The integer $e_0$ is the rank of $G$.

The finite case is just $e_0=0$. The rank counts the “infinite directions”; the cyclic prime-power factors are the torsion part.

Significance

Combined with Cayley’s theorem and the fact that $C_n$ embeds in $S_n$, this gives a complete classification of finite abelian groups in terms of integer partitions. The non-abelian classification is vastly harder (the Classification of Finite Simple Groups, 1980s).

See also Internal direct product for the detection mechanism.

Source Sections

• 11.2 Abelian Groups of Prime Power Order
• 11.3 The Classification Theorem of Finite Abelian Groups
• Worked problems: mth3003 weekly problems 11 (classification of order $72$; abelian converse to Lagrange).