Intersection of cyclic subgroups

In summary, the intersection of cyclic subgroups is the set of common elements between two or more cyclic subgroups within a larger group. It can be calculated by finding the common elements in the generators of the cyclic subgroups and can be empty if there are no common elements. The intersection has significance in group theory as it can help determine the structure and properties of a group and can also be used to prove theorems and solve problems. However, it may not always be a cyclic subgroup, depending on the elements in the intersection.
  • #1
Chen
977
1
This time I need a yes/no answer (but a definitive one!):
Suppose we have a group of finite order G, and two cyclic subgroups of G named H1 and H2. I know the intersection of H1 and H2 is also a subground of G, question is - is it also cyclic? And can I tell who is the creator of it, suppose I have the creators of H1 and H2?

Thanks,
Chen
 
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  • #2
Cyclic is easy. The intersection of H1 and H2 is a subgroup of H1. Subgroups of cyclic groups are cyclic.
 
  • #3
Thank you. :smile:
 

Related to Intersection of cyclic subgroups

1. What is the intersection of cyclic subgroups?

The intersection of cyclic subgroups refers to the set of elements that are common to two or more cyclic subgroups within a larger group.

2. How is the intersection of cyclic subgroups calculated?

The intersection of cyclic subgroups can be calculated by finding the common elements in the generators of the cyclic subgroups and then using those elements to generate a new subgroup.

3. Can the intersection of cyclic subgroups be empty?

Yes, it is possible for the intersection of cyclic subgroups to be empty if there are no common elements between the cyclic subgroups.

4. What is the significance of the intersection of cyclic subgroups in group theory?

The intersection of cyclic subgroups is important in group theory as it can help determine the structure and properties of a larger group. It can also be used to prove theorems and solve problems related to cyclic groups.

5. Is the intersection of cyclic subgroups always a cyclic subgroup?

No, the intersection of cyclic subgroups may not always be a cyclic subgroup. It depends on the elements in the intersection and their ability to generate a new subgroup.

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