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Direct product

 Direct Product:

If 𝐺 and 𝐻 are two groups (finite or infinite). Then the direct product of 𝐺 and 𝐻 is a 

new group, denoted by 𝐺 × 𝐻 and is defined by

𝐺 × 𝐻 = 𝑥, 𝑦 𝑥 ∈ 𝐺, 𝑦 ∈ 𝐻}.

The group operation defined is multiplication. Let 𝑎, 𝑏 ∈ 𝐺 and 𝑥, 𝑦 ∈ 𝐻, then

𝑎, 𝑏 . 𝑥, 𝑦 = (𝑎. 𝑥, 𝑏. 𝑦).

It is also called the external direct product.

Properties:

a) Identity: The direct product 𝐺 × 𝐻 has an identity element, namely {𝑒1, 𝑒2}, where 𝑒1 ∈ 𝐺 and 𝑒2 ∈ 𝐻.

b) Inverse: The inverse of each element (𝑥, 𝑦) ∈ 𝐺 × 𝐻 is (𝑥−1, 𝑦−1), where 𝑥−1 ∈ 𝐺 and 𝑦−1 ∈ 𝐻.

c) Associativity: 

The associative law holds in 𝐺 × 𝐻. That is, for 𝑥1, 𝑦1 , 𝑥2, 𝑦2 , (𝑥3, 𝑦3) ∈ 𝐺 × 𝐻

( 𝑥1, 𝑦1 . 𝑥2, 𝑦2). 𝑥3, 𝑦3= 𝑥1, 𝑦1. ( 𝑥2, 𝑦2.(𝑥3,𝑦3)).

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