What is the number of elements in the set {x^(13n) : n is a positive integer}?

In summary, if a cyclic group of order 15 has an element x such that the set {x^3, x^5, x^9} has exactly two elements, then the number of elements in the set {x^(13n) : n is a positive integer} is 3. This is because an element in a cyclic group can have an order that divides the order of the group, and the order of an element raised to a power can be found using the formula |x^k|=n/gcd(n,k). An example of a set that meets this condition is the group Z_{15} with elements 5 and 10 having order 3.
  • #1
yxgao
123
0
A cyclic group of order 15 has an element x such that the set {x^3, x^5, x^9} has exactly two elements. The number of elements in the set {x^(13n) : n is a positive integer} is :
3.

WHy is the answer 3? Thanks!
 
Mathematics news on Phys.org
  • #2
Let [tex]x[/tex] be an element of a cyclic group of order 15. If [tex]\{x^3,x^5,x^9\}[/tex] has exactly 2 elements, then one element must be the same as another. If [tex]|x|=3[/tex] then [tex]x^3=x^9[/tex]. If [tex]|x|=3[/tex] then [tex]|x^{13}|=3/gcd(3,13)}=3[/tex]. Hence [tex]|<x^{13}>|=3[/tex].

Doug
 
  • #3
Thanks so much for the explanation! However I haven't actually studied group theory before so there's some things I still am not sure about. Why did you assume |x| = 3? What formula did [tex]|x^{13}|=3/gcd(3,13)}=3[/tex] come from?
Can you give an example of a set that meets this condition?
Thanks!
 
  • #4
Originally posted by yxgao
Thanks so much for the explanation! However I haven't actually studied group theory before so there's some things I still am not sure about. Why did you assume |x| = 3? What formula did [tex]|x^{13}|=3/gcd(3,13)}=3[/tex] come from? Can you give an example of a set that meets this condition?
Thanks!

Your original post stated that there exists an element [tex]\inline{x}[/tex] such that [tex]\inline{\{x^3,x^5,x^9\}}[/tex] has exactly two elements. Thus I need to find two elements that are the same. There is a theorem that states if [tex]\inline{|x|=n}[/tex] then [tex]\inline{x^i=x^j}[/tex] if and only if [tex]\inline{n}[/tex] divides [tex]\inline{i-j}[/tex]. If [tex]\inline{x^3=x^5}[/tex] then [tex]\inline{|x|=2}[/tex] which is not possible in a cyclic group of order 15 because 2 does not divide 15. Likewise, [tex]\inline{x^5{\neq}x^9}[/tex]. However if [tex]\inline{|x|=3}[/tex] then [tex]\inline{x^3=x^9}[/tex] because 3 divides [tex]\inline{i-j=9-3=6}[/tex]. Also, a cyclic group of order 15 can have an element with order 3.

There is a theorem that states if [tex]\inline{|x|=n}[/tex] then [tex]\inline{|x^k|=n/gcd(n,k)}[/tex].

The group [tex]\inline{Z_{15}}[/tex] is the group of integers modulo 15 under addition. Both the element 5 and the element 10 have order 3.

Doug
 

What is a cyclic group of order 15?

A cyclic group of order 15 is a mathematical structure that consists of 15 elements and follows a specific set of rules. It is also known as a cyclic group of 15 elements or a group of 15th roots of unity.

What are the properties of a cyclic group of order 15?

A cyclic group of order 15 has the following properties:

  • It contains 15 elements.
  • It is generated by a single element.
  • It is commutative, meaning the order of multiplication does not matter.
  • It has a unique identity element, which is the element that does not change when multiplied by any other element.
  • Each element has an inverse, meaning there is another element that when multiplied with it, produces the identity element.

How do you find the generator of a cyclic group of order 15?

The generator of a cyclic group of order 15 can be found by raising any element to the power of 15. This will produce the identity element, and the element used to raise it to the power of 15 will be the generator.

How many subgroups does a cyclic group of order 15 have?

A cyclic group of order 15 has four subgroups, which are the trivial subgroup (containing only the identity element), the group itself, and two non-trivial subgroups of order 5 and 3 respectively.

What is the significance of a cyclic group of order 15?

A cyclic group of order 15 has many applications in mathematics and physics. It is used in number theory, cryptography, and the study of symmetry and patterns. It also has connections to the roots of unity, which are important in solving polynomial equations.

Similar threads

Replies
5
Views
839
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
7
Views
2K
Replies
2
Views
982
  • General Math
Replies
2
Views
953
Replies
4
Views
885
  • Precalculus Mathematics Homework Help
Replies
6
Views
648
  • General Math
2
Replies
68
Views
9K
Replies
1
Views
1K
Back
Top