Proof Group Homework: Cyclic if Has Order m & n Elements

In summary: The only other possibility would be that a is not a power of any other element in the group, but this contradicts our assumtions. So, it must be the case that G is cyclic.
  • #1
cragar
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3

Homework Statement


Let G be an ableian group of order mn, where m and n are relativiely prime. If G has
has an element of order m and an element of order n, G is cyclic.

The Attempt at a Solution


ok so we know there will be some element a that is in G such that
[itex] a^m=e [/itex] where e is the identity element. It seems that this would be enough to prove that their is a sub group generated by a. and this sub group is cyclic. if I start with the element a
all powers of a would need to be in their so it would be closed under the operation.
I guess we know its a group already. Let's say we have some power of a like x where
0<x<m we want to know if this has an inverse that is a power of a.
we know [itex] a^m=e [/itex] so if we have some arbitrary power of a [itex] a^x [/itex]
we want its inverse [itex] a^xa^p=e=a^{x+p}=a^m [/itex] so x+p=m so their is a cyclic subgroup
generated by a, Now we know that if we have a cyclic group all of its subgroups are cyclic.
I am slightly worried about the converse, is it always true if I have cyclic subgroup that the group is cyclic? But I guess i could just do the same argument with some element of the form
[itex] b^n=e [/itex] and then look at all the possible group operations. I guess I could try to find the generator for G.
 
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  • #3
mn, so I guess ab would be the generator of the group.
 
Last edited:
  • #4
cragar said:
mn, so I guess ab would be the generator of the group.

Well, you must argue that. Given its order and the size of the group, what's left?
 

1. What is a cyclic group?

A cyclic group is a type of mathematical group that is generated by a single element. This means that every element in the group can be expressed as a power or combination of that single element. Cyclic groups are commonly used in abstract algebra and have many applications in fields such as cryptography and physics.

2. What does it mean for a group to have order m & n elements?

The order of a group is the number of elements it contains. When a group has order m & n elements, it means that there are two distinct elements in the group, one with m elements and one with n elements. This can be seen in a group's multiplication table, where each element appears exactly once in each row and column.

3. How can you tell if a group is cyclic?

A group is cyclic if it can be generated by a single element. This means that every element in the group can be expressed as a power or combination of that single element. Additionally, if the group's order is equal to the order of the element generating it, then the group is cyclic.

4. What is the significance of the order of a cyclic group?

The order of a cyclic group is important because it determines the structure and behavior of the group. For example, the order of a cyclic group will determine the number of subgroups it contains and the possible orders of those subgroups. Additionally, the order of a cyclic group will also determine if it is isomorphic to another group.

5. Can a group have more than one generator?

Yes, a group can have multiple generators. This means that there are multiple elements in the group that can generate the entire group. However, every generator of a cyclic group will have the same order as the group itself.

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