Can (Z/10557Z)* be written as Cn1 x Cn2 x Cn3 with n1 dividing n2 dividing n3?

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In summary, an Abelian group is a mathematical structure with a set of elements and a commutative binary operation. They are significant in mathematics as a fundamental example of a group and have applications in various fields. To determine if a group is Abelian, the operation must be commutative. The order of an Abelian group is the number of elements and can be infinite.
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RVP91
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If I was to try to work this out I would use the Chinese Remainder Theorem and since 10557 = 3^3 . 17 . 23
end up with (Z/10557Z)* isomorphic to (Z/27Z)* x (Z/17Z)* x (Z/23Z)* isomorphic to C18 x C16 x C22 where Cn represents the Cyclic group order n.

How would I then write this as Cn1 x Cn2 x Cn3 s.t. n1 divides n2 divides n3?
 
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this is explained in section 17 of these free notes:

http://www.math.uga.edu/%7Eroy/844-2.pdf and probably in many other algebra books.
 
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1. What is an Abelian group?

An Abelian group is a mathematical structure that consists of a set of elements and a binary operation that combines any two elements in the set to produce a third element. Additionally, the operation must be associative, commutative, and have an identity element and inverse element for each element in the group.

2. What is the significance of Abelian groups in mathematics?

Abelian groups are important in mathematics because they provide a fundamental example of a group, which is a key concept in abstract algebra. They also have many applications in different areas of mathematics, such as number theory, geometry, and physics.

3. How do you determine if a group is Abelian?

To determine if a group is Abelian, you need to check if the group's operation is commutative. This means that if you switch the order of the elements in the operation, the result will be the same. If the operation is commutative, then the group is Abelian.

4. What is the order of an Abelian group?

The order of an Abelian group is the number of elements in the group. It is denoted by the symbol |G|, where G is the group. The order of an Abelian group must be a positive integer.

5. Can Abelian groups have infinite order?

Yes, Abelian groups can have infinite order. This means that the group has an infinite number of elements. Examples of infinite Abelian groups include the integers, rational numbers, and real numbers under addition.

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(Z/10557Z)* as Abelian Groups using Chinese Remainder Theorem
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