Fundamental Theorem of Finite Abelian Groups

Fundamental Theorem of Finite Abelian Groups states that: every finite Abelian group is a direct product of cyclic groups of prime power order. Moreover, the number of terms in the product and the orders of the cyclic groups are uniquely determined by the group.

Fundamental Theorem of Finite Abelian Groups

Every finite generated abelian group $G$ is isomorphic to a direct product of cyclic groups. i.e. $G\cong \Z_{p_1^{n_1}} \times \cdots \times \Z_{p_k^{n_k}}\times \Z^b$ where $p_i$ are prime numbers and $n_i$ are positive integers.

• We call $b$ the Betti number of $G$. A finite generated abelian group is finite if and only if its Betti number is zero.

Corollary: If $m | |G|$, then $G$ has a subgroup of order $m$.

Proof of the Fundamental Theorem of Finite Abelian Groups

Lemma 1

Let $G$ be a finite abelian group of order $p^n m$ where $p$ is prime that does not divide $m$. Then $G=H\times K$ where $H = \{x\in G|x^{p^n} = e\}$ and $K = \{x\in G|x^{m} = e\}$. Moreover, $|H| = p^n$ and $|K| = m$.

Lemma 2

Let $G$ be an abelian group of prime-power order and let $a$ be an element of maximum order in $G$. Then $G$ can be written in the form $G = \langle a\rangle \times K$.

Lemma 3

Let $G$ be an abelian group of prime-power order is an internal idrect product of cylic groups.

Lemma 4

Suppose $G$ is a finite abelian group of prime-power order. If $G = H_1 \times \cdots \times H_m$ and $G = K_1 \times \cdots \times K_n$ where $K_i$ and $H_i$ are nontrivial cyclic groups, with $|H_1| \ge |H_2| \ge \cdots \ge |H_m|$ and $|K_1| \ge |K_2| \ge \cdots \ge |K_n|$, then $m = n$ and $|H_i| = |K_i| \forall i$.

The isomorphism Classes of Abelian Groups

A group $G$ is called finitely generated if every element can be written down as a product of $g_i$s and their inverses.

Let $G$ be a finitely generated Abelian group. $\exists t\in \Z_{\ge0},$ the invariant factors $(k_i)_{i=1,\ldots,m}\in \N$ satisfying $k_1|k_2|\ldots|k_m$ such that $G\cong \Z^t\times \Z/k_1\Z\times \cdots\times \Z/k_m\Z$, $t, k_i$ are uniquely defined by $G$. We can also have form $G\cong \Z^t\times\prod_i \Z/p_i^{m_i}\Z$

• $|G| < \infty \implies t= 0, \Z^t=\{0\}$

Let $G$ be a finitely generated Abelian group.

• A set of elements $\{a_1,\ldots,a_n\}$ is called a basis of $G$. Every element of $G$ can be uniquely written down as $k_1a_1+\cdots + k_na_n, k_i \in \Z$
• $G$ with a basis is called a free Abelian you

All bases of a free Abelian group $G$ have the same number of elements. Every free Abelian group is isomorphic to $\Z^n$, that is, every free Abelian group $G$ has a rank of $rk(G) = n$

Let $N\subset L$ be a subgroup of a free Abelian group of rank $n$, then $N$ is also a free group of rank $\le n$. Then there exists a basis $\{e_1,\ldots, e_n\}$ of $L$ and positive integers $k_i$ such that $k_1e_1,\ldots,k_me_m$ is a basis of $N$ and $k_1|k_2|\ldots|k_m$

Define integer-valued elementary transformations $A$ any types of follows:

• $v_i,v_j$ are rows/columns, and $a\in \Z \implies v_i\mapsto v_i + av_j$
• any transposition of two rows/columns
• multiplying a row/column by $-1$

Recall: a diagonal $n\times m$ matrix is $\forall i\ne j, d_{ij} = 0$ or we can have form $diag(d_i,\ldots,d_{min(n,m)})$

Any integer-valued matrix $C=(c_{ij})\in Mat_{n\times m}(\Z)$ can be transformed via elementary transformations into a diagonal matrix $diag(d_1,\ldots,d_p)$ where $d_i \ge 0$ and $d_1|d_2|\ldots|d_p$

$G$ is a finitely generated Abelian group $\iff \exists$ a surjective homomorphism $\psi:\Z^n \to G$

A finite abelian group $G$ is cyclic $\iff exp(G) = |G|$

$(\Z/n\Z)^{\times}$ is cyclic when $n = 2,4,p^k, 2p^k$