Solved: Group Theory Problem: Cyclic Group of Order 15

In summary, the conversation discusses a group theory problem involving a cyclic group of order 15 and an element x. It is determined that x^4 = 1 and therefore the set {x^{13n} : n is a positive integer} has only one element, leading to answer a) being the only possible option. The use of Lagrange's Theorem is also mentioned in solving the problem.
  • #1
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[SOLVED] group theory problem

Homework Statement


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

a)3
b)5


Homework Equations





The Attempt at a Solution


From the given information, we know that x^6 = 1 or x^4 = 1. In the first case, either answer is possible. In the second case, only answer a is possible. Anyway, do we know enough to decide which case this is or is there a different way to do the problem?
 
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  • #2
ehrenfest said:
From the given information, we know that x^6 = 1 or x^4 = 1.
True. But why stop there? (I'm assuming that you have a valid justification for this assertion)

In the first case, either answer is possible. In the second case, only answer a is possible.
I can't guess at your reasoning for either conclusion; would you show your work?
 
  • #3
if x^4 = 1, you can conclude what x is on what you know about the size of the group. the answer then follows.
 
  • #4
Oh. I see. By Lagrange's Theorem, x^4=1 implies x=1 which implies there is exactly one element in the set. x^6 implies x^3=3 again by Lagrange and the fact that there is more than one element in the set. The answer a) is immediate since gcd(13,3)=1. Thanks.
 

What is a cyclic group of order 15?

A cyclic group of order 15 is a mathematical structure that consists of a set of elements and a binary operation, where the operation follows certain rules. In this case, the group has 15 elements and is generated by a single element called a generator.

How do you determine the elements of a cyclic group of order 15?

To determine the elements of a cyclic group of order 15, we first need to find the generator of the group. This is done by finding an element that, when multiplied by itself a certain number of times, will generate all the elements of the group. In this case, the generator can be any number between 1 and 14 that is relatively prime to 15, such as 2, 4, 7, or 11. The elements of the group will then be the powers of the generator, starting from 1 until we reach 15.

What is the identity element of a cyclic group of order 15?

The identity element of a cyclic group of order 15 is the element that, when combined with any other element in the group, will yield that same element. In other words, the identity element is the element that has no effect when combined with another element. In this case, the identity element is 1, as 1 multiplied by any other element will result in that same element.

What is the inverse element in a cyclic group of order 15?

The inverse element in a cyclic group of order 15 is the element that, when combined with another element, will yield the identity element. In other words, the inverse element is the element that, when multiplied by another element, will result in 1. In this case, the inverse of an element x in the group is denoted as x^-1 and can be found by raising x to the power of 14, as 15-1=14.

How is a cyclic group of order 15 different from other groups?

A cyclic group of order 15 is different from other groups in that it is generated by a single element, whereas other groups may have multiple generators. It is also different in that it has a finite number of elements, whereas other groups may have an infinite number of elements. Additionally, each element in a cyclic group of order 15 has a unique inverse, which may not be the case in other groups.

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