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created: 2021-12-06 10:48:24
modified: 2022-01-10 04:13:03
and groups.
is the direct product of these groups. It is a group if we define the product of its elements this way:
With set notation:
Simplified notation
The groups are and and their product is .
Element factorization
Every element in the product group can be factorized the following way:
Useful identification
We can use the following identifications: and . With this, the element can be written simplified:
It is easy to see that any and commutes.
Intuitively, and are two independently operating groups, acting on two independent degrees of freedom. This is why they are commutative.