Abstract algebra cyclic subgroups

In summary, the conversation discusses determining the number of cyclic subgroups of order 10 in a group with eight elements of that same order. The Euler phi function is mentioned as a way to calculate the number of elements of order 10 needed in the group. It is concluded that there can only be two cyclic subgroups of order 10 in the group, as they cannot share any elements of that order.
  • #1
tyrannosaurus
37
0

Homework Statement


Suppose that G is a group with exactly eight elements of order 10. How many cyclic subgroups of order 10 does G have?

Homework Equations





The Attempt at a Solution


I really don't have a clue how to solve this, any help would be greatly appreciated.
 
Physics news on Phys.org
  • #2
If you have one cyclic subgroup of order 10, how many elements of order 10 does that force G to have?
 
  • #3
you would need four elements of order 10 (by the Euler phi function). does that mean there would be two cyclicsubgroups of order2?
 
  • #4
tyrannosaurus said:
you would need four elements of order 10 (by the Euler phi function). does that mean there would be two cyclicsubgroups of order2?

You tell me. You definitely need more than one. Can two cyclic subgroups of order 10 share any elements of order 10?
 
  • #5
there would only be 2 cyclic subgroups of order 10 because non of the subgroups can share an order 10 element because if they did share an element in common, that element would generate both groups, so the two groups would be the same. So this means that no two cyclic subgroups of order 10 can share an element of order 10.
Thanks a lot for the help, this made sooooooo much more sense now.
 

1. What is a cyclic subgroup in abstract algebra?

A cyclic subgroup in abstract algebra is a subset of a group that is generated by a single element, also known as a generator. This means that the subgroup contains all the powers and inverses of the generator, and it is closed under the group operation.

2. How do you determine if a subgroup is cyclic?

A subgroup is cyclic if and only if it can be generated by a single element. This means that every element in the subgroup can be written as a power of the generator. To determine if a subgroup is cyclic, you can check if there is an element in the subgroup that generates all the other elements.

3. What is the order of a cyclic subgroup?

The order of a cyclic subgroup is equal to the order of its generator. For example, if the generator has an order of 5, then the cyclic subgroup will also have an order of 5. This is because the subgroup will contain the identity element, the generator, and its powers up to the order.

4. Can a cyclic subgroup have more than one generator?

No, a cyclic subgroup can only have one generator. This is because the definition of a cyclic subgroup is a subset of a group that is generated by a single element. If a subgroup has more than one generator, then it is not considered a cyclic subgroup.

5. What is the relationship between cyclic subgroups and cyclic groups?

A cyclic subgroup is a subset of a group, while a cyclic group is a group where every element can be generated by a single element. This means that every cyclic subgroup is a cyclic group, but not every cyclic group is a cyclic subgroup. Cyclic subgroups are used to study the structure of cyclic groups.

Similar threads

  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
4K
  • Calculus and Beyond Homework Help
Replies
5
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
3K
  • Calculus and Beyond Homework Help
Replies
5
Views
2K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
3K
Back
Top