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\$