How are the orders in a cyclic group related?

[Genady]

G is a cyclic group of order n generated by an element g, G = {e, g, g², ..., g^n-1). It can be also generated by its other element h = g^k, G = {e, h, h², ..., h^n-1}. How are the k and n related?

[KJW]

24 minutes ago, Genady said:

It can be also generated by its other element h = g^k, G = {e, h, h², ..., h^n-1}.

No, the group generated by another element is a subgroup of G, not necessarily G itself.

 

24 minutes ago, Genady said:

How are the k and n related?

k is a factor of n (or perhaps a multiple of a factor of n).

 

[Genady]

2 minutes ago, KJW said:

No, the group generated by another element is a subgroup of G, not necessarily G itself.

 

k is a factor of n (or perhaps a multiple of a factor of n).

 

"It can be also generated by its other element" is a given.

[KJW]

2 minutes ago, Genady said:

"It can be also generated by its other element" is a given.

Then k and n are relatively prime.

 

[Genady]

41 minutes ago, KJW said:

Then k and n are relatively prime.

 

It is also sufficient, right?

Edited December 27, 2024 by Genady