Order of elements in finite abelian groups

In summary, the conversation discusses how to prove that in a finite and abelian group G with the least common multiple m of its element's order, there exists an element of order m. This is due to the fact that the order of the product of two elements is not necessarily the least common multiple of their orders, and the example of a nonabelian group shows that abelianity is necessary for this statement to hold. The structure theorem for finite abelian groups can be used to prove this statement.
  • #1
jacobrhcp
169
0
prove that if G is a finite and abelian group and m is the least common multiple of the order of it's element, that there is an element of order m.

My idea:

if ai are the elements of G, the order of a1*a2 is lcm(a1,a2) and the result follows directly when applied to all ai... but why is this correct and why is this only for abelian groups?
 
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  • #2
The order of a1*a2 is not lcm(o(a1),o(a2)), e.g. take a nonidentity element and its inverse: the order of their product is 1, but the lcm of their orders is >1.

And yes, abelian is necessary here. (Try to find an example of a finite nonabelian group in which this is not true.)
 
  • #3
I did that, that was the next question in the book =P,
D3 is a finite nonabelian group, in which the elements have order 1,2, or 3. The least common multiple of these is 6 and the result is not true.
 
  • #4
Yup, that works. For the original problem try looking at the structure theorem for finite abelian groups.
 

1. What is the order of elements in a finite abelian group?

The order of an element in a finite abelian group is the smallest positive integer n such that the element raised to the nth power is equal to the identity element.

2. How do you determine the order of an element in a finite abelian group?

To determine the order of an element in a finite abelian group, you can raise the element to different powers until you reach the identity element. The smallest power that results in the identity element is the order of the element.

3. Can an element in a finite abelian group have multiple orders?

No, an element in a finite abelian group can only have one order. This is because the order of an element is defined as the smallest positive integer n that results in the identity element when the element is raised to the nth power.

4. How does the order of an element affect the structure of a finite abelian group?

The order of elements in a finite abelian group plays a crucial role in determining the structure of the group. It helps to classify the group into different types, such as cyclic, dihedral, or direct product of cyclic groups.

5. Are there any patterns or relationships between the orders of elements in a finite abelian group?

Yes, there are patterns and relationships between the orders of elements in a finite abelian group. For example, if the group is cyclic, all elements will have the same order. In general, the orders of elements must be divisors of the order of the group. Additionally, the orders of elements may also be related to the prime factorization of the group's order.

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