The main goal of this post is to summarise a few fundamental statements for working with groups of type . It is going to be very technical and maybe difficult to learn for the first time you see it, but give it a chance and you will see that it actually makes sense.

## Sources:

Drapal part 2.10

## Theory:

First, let us recall some consequences of the Chinese remainder theorem: Let , such that ’s are pairwise coprime. Then

- .
- .

Except of the Chinese remainder theorem, we have two more new (rather technical) propositions dealing with properties of groups , namely with the group structure and elements of order :

Let be a prime number and let . Then

- Set . Then is a subgroup of of order .
- An element is of order in .
- If , then and are also of order in .

Again, let be a prime number and let . Then

- If , then .
- If , then .
- Finally, if is odd, then is a cyclic group of order , i.e., .