Direct product of groups

The (external) direct product is the categorical product in $𝖦 𝗋 𝗉$ . Given two groups $𝐴, 𝐵$ , their product $𝐴 \times 𝐵$ is their Cartesian product with the group operation $(\cdot)$ such that #m/def/group

(𝑎 1, 𝑏 1) \cdot (𝑎 2, 𝑏 2) = (𝑎 1 𝑎 2, 𝑏 1 𝑏 2)

for any $𝑎 1, 𝑎 2 \in 𝐴$ and $𝑏 1, 𝑏 2 \in 𝑉$ . It follows that $𝑒 𝐴 \times 𝐵 = (𝑒 𝐴, 𝑒 𝐵)$ and $(𝑎, 𝑏) - 1 = (𝑎 - 1, 𝑏 - 1)$ . This generalized to arbitrarily large products

𝐺 = \prod 𝑖 \in 𝐼 𝐺 𝑖 𝑈

The projections $𝜋 𝑖 : 𝐺 ↠ 𝐺 𝑖$ are split epic.

Internal direct product

Noting ^P3, a related internal construction occurs when there exist normal subgroups $𝑁, 𝑀 ⊴ 𝐺$ such that $𝑁 \cap 𝑀 = {𝑒}$ and $𝑁 𝑀 = 𝐺$ . #m/def/group This motivates generalisation to the Semidirect product (both external and internal), where only one group need be normal.

Properties

If $𝟙$ is the trivial group, $𝐺 \times 𝟙 ≅ 𝐺 ≅ 𝟙 \times 𝐺$ P1
Clearly $| 𝐴 \times 𝐵 | = | 𝐴 | | 𝐵 |$ .
$𝐴 ≅ {(𝑎, 𝑒 𝐵) : 𝑎 \in 𝐴} ⊴ 𝐴 \times 𝐵$
$𝐴 ≅ (𝐴 \times 𝐵) / ({𝑒 𝐴} \times 𝐵)$ . Usually this is stated as $𝐴 = (𝐴 \times 𝐵) / 𝐵$ . However it is not generally true that given $𝐻 ⊴ 𝐺$ we have $𝐺 ≅ 𝐻 \times (𝐺 / 𝐻)$ .

#state/tidy | #lang/en | #SemBr