Finite Abelian Groups

Subjects: Group Theory
Links: Fourier Analysis on finite abelian groups, Groups, Direct Product of Groups, Cyclic Groups

Th: Every finite abelian group $G$ it can be decomposed as the product of cyclic groups: $$G \cong C(m_1)\times \dots \times C(m_n) $$where each $C(m_i)$ is the cyclic group of order $m_i$, and $m_i\mid m_{i+1}$ for each $i=1,\dots ,n$.

Cor: If $G$ is a finite abelian group and $k$ a divisor of $|G|$, then there's a subgroup of order $k$.

Def: If $p$ is a prime, then an abelian group $G$ is a $p$-primary if for each $a\in G$, then there is $n\ge 1$ with $p^{n}a=0$. If $G$ is an abelian group, the its $p$-primary component is $$G_p:= { a \in G \mid p^n a = 0 \text{ for some } n \ge 1}$$
Lemma:

Primary Decomposition Theorem:

• Every finite abelian group $G$ is the direct product of its $p$-primary components: $$G \cong \prod_{p} G_p$$
• Two finite abelian groups $G$ and $G^{\prime }$ are isomorphic iff $G_p\cong G_p^{\prime }$ for every prime.