Cyclic Subgroups in Symmetric and Cyclic Groups

  • Thread starter chibulls59
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In summary, there are multiple cyclic subgroups in the group H x K, where K is a cyclic group with 2 elements and H is a symmetric group with 6 elements. The elements of H x K all generate cyclic subgroups, some of which may be the same. The task is to determine the total number of unique cyclic subgroups in H x K.
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chibulls59
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Suppose K= < x > is a cyclic group with 2 elements and H= S3 is symmetric group with 6 elements. Find all different cyclic subgroups of G= H x K.

Now since K is generated by x with 2 elements, I have K= {1,x} and H= {1, (12), (13), (23), (123), (132)}

What I am confused about is finding cyclic subgroups of H x K. Am I supposed to be checking each element of H x K and seeing if it can generate the whole group?
 
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  • #2
All elements of the group generate a cyclic group. Some of them generate the same cyclic group. You are just supposed to figure out how many there are. The whole group isn't cyclic.
 

Related to Cyclic Subgroups in Symmetric and Cyclic Groups

1. What is a cyclic subgroup?

A cyclic subgroup is a subset of a group that is generated by a single element of that group. This means that by repeatedly applying the group's operation to the generator element, all other elements in the subgroup can be obtained.

2. How do you find cyclic subgroups?

To find cyclic subgroups, you need to first identify the generator element of the group. This can be done by testing each element of the group to see if it generates a cyclic subgroup. Once the generator element is found, the cyclic subgroup can be formed by repeatedly applying the group's operation to the generator element.

3. Can a group have multiple cyclic subgroups?

Yes, a group can have multiple cyclic subgroups. In fact, every element in a cyclic group is a generator for a cyclic subgroup. This means that if a group has n elements, it can have n cyclic subgroups.

4. What is the order of a cyclic subgroup?

The order of a cyclic subgroup is the number of elements in that subgroup. This is equal to the order of the generator element. For example, if the generator element has an order of 3, then the cyclic subgroup will also have an order of 3.

5. How are cyclic subgroups used in mathematics?

Cyclic subgroups are used in various branches of mathematics, such as group theory, abstract algebra, and number theory. They help to understand the structure of a group and can be used to classify and solve problems related to groups. They also have applications in cryptography and coding theory.

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