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.