The Canonical Injections of Weak Direct Products of Groups

The Canonical Injections of Weak Direct Products of Groups

Recall from The Weak Direct Product of an Arbitrary Collection of Groups page that if $\{ G_i : i \in I \}$ is an arbitrary collection of groups the the weak direct product of the groups $\{ G_i : i \in I \}$ is the set:

(1)
\begin{align} \quad \prod_{i \in I}^{\mathrm{weak}} G_i = \left \{ f : I \to \bigcup_{i \in I} G_i \: \biggr | \: f(i) \in G_i \: \forall i \in I, \: f(i) = e_{G_i} \: \mathrm{for \: all \: but \: finitely \: many \:} i \right \} \end{align}

with the operation of pointwise product, defined for all $f, g \in \prod_{i \in I}^{\mathrm{weak}} G_i$ by $(fg)(i) = f(i)g(i)$ for all $i \in I$. We proved that $\prod_{i \in I}^{\mathrm{weak}} G_i$ is a normal subgroup of $\prod_{i \in I} G_i$.

We will now define some important functions related to the weak direct product of groups.

 Definition: Let $\{ G_i : i \in I \}$ be an arbitrary collection of groups. The Canonical Injection of $G_j$ into $\prod_{i \in I}^{\mathrm{weak}}$ is the map $\iota_j : G_j \to \prod_{i \in I}^{\mathrm{weak}}$ defined for all $g \in G$ by $\iota_j(g)$ to be the function defined for all $i \in I$ by $[\iota_j(g)](i) = \begin{Bmatrix} g & \mathrm{if} \: i = j \\ e_{G_i} & \mathrm{if} \: i \neq j \end{Bmatrix}$.

In other words, each $\iota_j(g)$ is the function on $I$ that maps every $i \in I$ to $e_{G_i}$ with the exception that $\iota_j(g)$ maps $j$ to $g$.

The following proposition tells us that each canonical injection $\iota_j$ is a monomorphism from $G_j$ to $\prod_{i \in I}^{\mathrm{weak}} G_i$.

 Proposition 1: Let $\{ G_i : i \in I \}$ be an arbitrary collection of groups. Then for each $j \in I$, the canonical injection $\iota_j : G_j \to \prod_{i \in I}^{\mathrm{weak}} G_i$ is a monomorphism.
• Proof: Let $j \in I$ and let $g, g' \in G_J$. Then:
(2)
\begin{align} \quad [\iota_j(g)](i) &= \begin{Bmatrix} g & \mathrm{if} \: i = j \\ e_{G_i} & \mathrm{if} \: i \neq j \end{Bmatrix} \\ \quad [\iota_j(g')](i) &= \begin{Bmatrix} g' & \mathrm{if} \: i = j \\ e_{G_i} & \mathrm{if} \: i \neq j \end{Bmatrix} \end{align}
• Therefore:
(3)
\begin{align} \quad [\iota_j(g) \iota_j(g')](i) &= \begin{Bmatrix} gg' & \mathrm{if} \: i = j \\ e_{G_i} & \mathrm{if} \: i \neq j \end{Bmatrix} = \iota_j(gg') \end{align}
• So indeed, for all $g, g' \in G_j$ we have that $\iota_j(gg') = \iota_j(g) \iota_j(g)$, so $\iota_j$ is a homomorphism.
• Now let $g, g' \in G_j$ and suppose that $\iota_j(g) = \iota_j(g')$. Then $[\iota_j(g)](j) = [\iota_j(g')](j)$, or equivalently, $g = g' 4]]. So [[$ \iota_j$is injective. • Thus$\iota_j : G_j \to \prod_{i \in I}^{\mathrm{weak}} G_i$is a monomorphism.$\blacksquare$ Proposition 2: Let$\{ G_i : i \in I \}$be an arbitrary collection of groups. Then for each$j \in J$,$\iota_j(G_j)$is a normal subgroup of$\prod_{i \in I} G_i$. • Proof: By Proposition 1,$\iota_j$is a homomorphism of$G_j$to$\prod_{i \in I}^{\mathrm{weak}} G_i$and so$\iota_j(G_j)$is a subgroup of$\prod_{i \in I}^{\mathrm{weak}} G_i$. Furthermore,$\prod_{i \in I}^{\mathrm{weak}} G_i$is a subgroup of$\prod_{i \in I} G_i$. So$\iota_j(G_j)$is a subgroup of$\prod_{i \in I} G_i$. • Let$G = \prod_{i \in I} G_i$and let$H = \iota_j(G_j)$. We aim to show that for all$g \in G$that$gHg^{-1} \subseteq H$. • Let$g \in G$and let$h \in H$. Since$h \in H$there exists an$a \in G_j$such that$\iota_j(a) = h$. So$h(j) = a$and$h(i) = e_{G_i}$for all$i \in I \setminus \{ j \}$. • So if$i = jwe have that: (4) \begin{align} \quad [ghg^{-1}](i) = g(j)h(j)g^{-1}(j) = g(j)ag^{-1}(j) \end{align} • And ifi \in I \setminus \{ j \}we have that: (5) \begin{align} \quad [ghg^{-1}](i) = g(i)h(i)g^{-1}(i) = g(i)e_{G_i}g^{-1}(i) = g(i)g^{-1}(i) = e_{G_i} \end{align} • Letghg^{-1} = \iota_j(g(j)ag^{-1}(j)) \in \iota_j(G_j)$. Thus$gHg^{-1} \subseteq H$which shows that$H = \iota_j(G_j)$is a normal subgroup of$\prod_{i \in I} G_i$.$\blacksquare\$