Cyclic subgroups of an Abelian group

In summary, if G is an Abelian group containing cyclic subgroups of orders 4 and 6, it must also contain cyclic subgroups of orders 1, 2, 3, and 12. This is because a cyclic group of order n has cyclic subgroups with orders corresponding to all of n's divisors, and since 4 and 6 are divisors of 12, so must 12 be a divisor of n.
  • #1
dancavallaro
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Homework Statement


If G is an Abelian group and contains cyclic subgroups of orders 4 and 6, what other sizes of cyclic subgroups must G contain?


Homework Equations


A cyclic group of order n has cyclic subgroups with orders corresponding to all of n's divisors.


The Attempt at a Solution


I know that a cyclic group of order n has cyclic subgroups with orders corresponding to all of n's divisors, so I'm inclined to say that G must have cyclic subgroups with orders 1, 2, and 3. But I also have a hunch that the subgroups of orders 4 and 6 combine in some way, so maybe there's also a cyclic subgroup of order lcm(4,6) = 12? Am I on the right track here?
 
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  • #2
Yes you are. Since 4 and 6 divide n, so must 12. (Think of the prime factorization of n).
 

1. What is a cyclic subgroup of an Abelian group?

A cyclic subgroup of an Abelian group is a subgroup that is generated by a single element. This means that all the elements in the subgroup can be obtained by repeatedly applying the group operation to the generator element. In other words, the subgroup "cycles" through the generator element.

2. How do you determine the order of a cyclic subgroup?

The order of a cyclic subgroup is equal to the order of its generator element. This means that if the generator element has order n, the cyclic subgroup will have n elements.

3. Are all subgroups of an Abelian group cyclic?

No, not all subgroups of an Abelian group are cyclic. In fact, only a subset of subgroups of an Abelian group are cyclic, namely those that are generated by a single element.

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

No, a cyclic subgroup can only have one generator. This is because the generator element is the unique element that generates all the other elements in the subgroup through repeated application of the group operation.

5. How are cyclic subgroups related to the concept of symmetry?

Cyclic subgroups are closely related to the concept of symmetry, as they represent a type of symmetry known as rotational symmetry. This is because the generator element "rotates" through the other elements in the subgroup, creating a cyclic pattern.

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