Direct products

Section 1.23 Direct products

We begin with a characterization of groups that are isomorphic to a direct product.

Theorem 1.23.1. Characterization of direct products.

A group \(G\) is isomorphic to a direct product \(\prod_{i=1}^r G_i\) if and only if the conditions below are satisfied.

For all \(1\leq i\leq r\) there is a normal subgroup \(N_i\normalin G\) and an isomorphism \(\phi_i\colon G_i\rightarrow N_i\text{.}\)
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The normal subgroups \(N_i\) satisfy

\begin{align} \prod_{i=1}^rN_i \amp = G\tag{1.23.1}\\ N_i\cap \prod_{j\ne i}N_j\amp =\{e\}, \text{ for all } 1\leq i\leq r \amp \text{.}\tag{1.23.2} \end{align}

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Proof.

The forward implication is relatively straightforward. Given \(G=\prod_{i\in I}G_i\text{,}\) where \(I=\{1,2,\dots, r\}\text{,}\) define

\begin{equation*} N_i=\{(g_j)_{j\in I}\mid g_j=e_{G_j} \text{ for all } j\ne i\}\text{.} \end{equation*}

It is easy to see that the collection \(N_1,N_2,\dots, N_r\) satisfy conditions (i) and (ii).

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For the reverse implication, given normal subgroups \(N_1,N_2,\dots, N_r\) satisfying conditions (i) and (ii), define

\begin{align*} \phi\colon \prod_{i=1}^rG_i \amp \rightarrow G\\ (g_1,g_2,\dots, g_n) \amp \mapsto \phi_1(g_1)\phi_2(g_2)\cdots \phi(g_r)\text{.} \end{align*}

We claim \(\phi\) is an isomorphism. Once we know \(\phi\) is a homomorphism, it is not dificult to show that it is an isomorphism. Indeed, surjectivity follows from (1.23.1); and injectivity follows from (1.23.2), which implies \(\ker\phi=\{e\}\text{.}\) That \(\phi\) is a homomorphism follow from the fact that if elements from different subgroups \(N_i\) commute with one another. We leave the details to the reader.

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Definition 1.23.2. Interal direct product.

Given a group \(G\) and normal subgroups \(N_1, N_2, \dots, N_r\) satisfying (i) and (ii) of Theorem 1.23.1, we say that \(G\) is the internal direct product of the subgroups \(N_1, N_2, \dots, N_r\text{.}\)

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Corollary 1.23.3. Internal direct products of Sylow subgroups.

Let \(G\) be a finite group of cardinality \(n> 1\text{,}\) and let \(p_1,p_2,\dots, p_r\) be the distinct prime divisors of \(G\text{.}\) The following conditions are equivalent.

\(G\) is an internal direct product of its Sylow subgroups.
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\(n_{p_i}(G)=1\) for all \(1\leq i\leq r\text{.}\)
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Corollary 1.23.4. Sylow theorems for abelian groups.

Let \(G\) be a finite abelian group of cardinality \(n> 1\text{,}\) and let \(p_1,p_2,\dots, p_r\) be the prime divisors of \(n\text{.}\)

For each \(1\leq i\leq r\) there is a unique \(p_i\)-Sylow subgroup of \(G\text{,}\) \(N_i\text{,}\) and

\begin{equation*} N_i=\{g\in G\mid \ord g=p_i^{n_i} \text{ for some } n_i\}\text{.} \end{equation*}

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\(G\) is the internal direct product of the \(N_i\text{.}\) In particular, we have

\begin{equation*} G\cong N_1\times N_2\times \cdots \times N_r\text{.} \end{equation*}

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Definition 1.23.5. Torsion subgroups.

Let \(G\) be an abelian group. For all prime numbers \(p\text{,}\) the \(p\)-torsion subgroup of \(G\text{,}\) denoted \(G(p)\) is defined as the set of all elements of \(G\) whose order is a power of \(p\text{:}\) i.e.,

\begin{equation*} G(p)=\{g\in G\mid \ord g=p^n \text{ for some } n\in \Z\}\text{.} \end{equation*}

The torsion subgroup of \(G\) is the set of all elements of \(G\) of finite order.

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Theorem 1.23.6. Structure of finite abelian groups: elementary divisors.

Let \(G\) be a finite abelian group of cardinality \(n> 1\text{,}\) let \(p_1,p_2,\dots, p_r\) be the distinct prime divisors of \(n\text{.}\) Assume \(n\) factors as

\begin{equation*} n=\prod_{i=1}^rp_i^{n_i}\text{.} \end{equation*}

\(G\cong \prod_{i=1}^r P_i\text{,}\) where \(P_i\) is an abelian group of cardinality \(p_i^{n_i}\text{.}\)
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For all \(1\leq i\leq r\) there are is a unique sequence of integers \((m_{i,1},m_{i,2},\dots, m_{i,r_i})\) satisfying

\begin{align*} 0< m_{i,1} \leq m_{i,2}\amp\leq \cdots \leq m_{i,r_i}\amp \\ m_{i,1}+m_{i,2}\amp+\cdots m_{i,r_i} =n_i\\ P_i\amp\cong \prod_{j=1}^{r_i}\Z/p_i^{m_{i,j}}\Z \amp \text{.} \end{align*}

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Proof.

In Math 331-2 we will prove a more general result, called the structure theorem for finitely generated modules over PIDs that implies this theorem.

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