Is the Product of A and An in a Cyclic Group of Order n Outside the Group?

[yicong2011]

Given a cyclic group of order n, with all its elements in the form :

A, A², A³, ..., Aⁿ

where A is an arbitrary element of the group.



According to the definition of group,

"The product of two arbitrary elements A and B of the group must be an element C of the group",

That is to say,

AB = C = an element of G


I just wonder that within a cyclic group, the product of element A and Aⁿ should be Aⁿ⁺¹, yet Aⁿ⁺¹ is not one of the element of a cyclic group of order n.
(since, all elements within a cyclic group of order n should has the form :

A, A², A³, ..., Aⁿ

where A is an arbitrary element of the group.

)

 

[Stephen Tashi]

In a finite cyclic group of order n and generated by the element $A$
$A^{n+1} =$ the identity element of the group

 

[spamiam]

In a finite cyclic group of order n and generated by the element $A$
$A^{n+1} =$ the identity element of the group

Don't you mean Aⁿ = 1? So then Aⁿ⁺¹ = A.

 

[Stephen Tashi]

Don't you mean Aⁿ = 1? So then Aⁿ⁺¹ = A.

You're right.

 

[blue_raver22]

This is indeed a valid observation. The product of two elements in a cyclic group should result in another element within that group. In this case, A and An are both elements of the group, but their product An+1 is not. This can be seen as a limitation of the definition of a cyclic group of order n, as it does not include all possible products of elements within the group. However, this does not necessarily mean that the group is not cyclic or that there is a problem with the definition. It simply means that some products may not be included in the set of elements of the group. This could be further explored and studied in the context of the specific group and its properties.