Abelian Groups of Order $2100$: Elements of Order $210$

In summary, there are four abelian groups of order $2100$: $\mathbb Z_{10} \times \mathbb{Z}_{210}, \mathbb Z_{2} \times \mathbb{Z}_{1050}, \mathbb Z_{5} \times \mathbb{Z}_{420},$ and $\mathbb{Z}_{2100}$. An element of order 210 in each group can be found by taking the identity element and adding 5 to itself 210 times, resulting in (0,5) for $\mathbb Z_{2} \times \mathbb{Z}_{1050}$. The element "e" order is the smallest number "n" such that $
  • #1
Megus1
4
0
Find all the abelian groups of order $2100.$ For each group, give an example of an element of order $210.$

$2100 = 2^2 \cdot 3 \cdot 5^2 \cdot 7,$ then

$G_1 = \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb Z_3 \times \mathbb{Z}_5 \times \mathbb{Z}_5 \times \mathbb{Z}_7 \cong \mathbb Z_{10} \times \mathbb{Z}_{210}$

$G_2 = \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_{5^2} \times \mathbb{Z}_7 \cong \mathbb Z_{2} \times \mathbb{Z}_{1050}$

$G_3 = \mathbb{Z}_{2^2} \times \mathbb{Z}_3 \times \mathbb{Z}_5 \times \mathbb{Z}_5 \times \mathbb{Z}_7 \cong \mathbb Z_{5} \times \mathbb{Z}_{420}$

$G_4 = \mathbb{Z}_{2^2} \times \mathbb{Z}_3 \times \mathbb{Z}_{5^2} \times \mathbb{Z}_7 \cong \mathbb{Z}_{2100}$

How do I find the elements of order 210? I don't get it well, for example for $\mathbb Z_{1050},$ an element of order 210 is $\dfrac{1050}{210}=5,$ but why?
 
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  • #2
Megus said:
Find all the abelian groups of order $2100.$ For each group, give an example of an element of order $210.$

$2100 = 2^2 \cdot 3 \cdot 5^2 \cdot 7,$ then

$G_1 = \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb Z_3 \times \mathbb{Z}_5 \times \mathbb{Z}_5 \times \mathbb{Z}_7 \cong \mathbb Z_{10} \times \mathbb{Z}_{210}$

$G_2 = \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_{5^2} \times \mathbb{Z}_7 \cong \mathbb Z_{2} \times \mathbb{Z}_{1050}$

$G_3 = \mathbb{Z}_{2^2} \times \mathbb{Z}_3 \times \mathbb{Z}_5 \times \mathbb{Z}_5 \times \mathbb{Z}_7 \cong \mathbb Z_{5} \times \mathbb{Z}_{420}$

$G_4 = \mathbb{Z}_{2^2} \times \mathbb{Z}_3 \times \mathbb{Z}_{5^2} \times \mathbb{Z}_7 \cong \mathbb{Z}_{2100}$

How do I find the elements of order 210? I don't get it well, for example for $\mathbb Z_{1050},$ an element of order 210 is $\dfrac{1050}{210}=5,$ but why?

what is the element "e" order ? it is the least number "n" such theat ne = Identity or if the operation is product e^n = I
when we add 5 to itself 210 we will get 1050 which is equal 0 the identity of $\mathbb Z_{1050},$ . using that we can find an element in
$\mathbb Z_{2} \times \mathbb{Z}_{1050}$
with order 210 which is (0,5)
 

FAQ: Abelian Groups of Order $2100$: Elements of Order $210$

What is an Abelian Group?

An Abelian group is a mathematical structure that consists of a set of elements and a binary operation that satisfies the properties of commutativity, associativity, and the existence of an identity element. In simpler terms, it is a group in which the order of operations does not matter, and all elements can be added/subtracted/multiplied/divided in any order without changing the result.

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 |G|, where G is the group. In this case, the order of the Abelian group is 2100.

What is the significance of an Abelian Group of Order 2100?

An Abelian group of order 2100 has 2100 elements. This means that it has a large number of elements, making it a useful tool in many mathematical applications. Furthermore, the number 2100 has many divisors, which allows for the formation of various subgroups within the Abelian group.

What does it mean for an element to have an order of 210?

The order of an element in an Abelian group is the smallest positive integer n such that the element raised to the power of n is equal to the identity element. In this case, an element with an order of 210 means that when it is multiplied by itself 210 times, it will result in the identity element.

What are the elements of order 210 in this Abelian Group?

The elements of order 210 are the elements in the group that, when multiplied by themselves 210 times, result in the identity element. In this Abelian group, there are 10 elements of order 210, and they are all distinct from each other.

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