Cyclic Subgroup H=<9> of Z30: List and Find Elements

[tasha10]

1. (a) List all elements in H=<9>, viewed as a cyclic subgroup of Z30
(b) Find all z in H such that H=<z>





I'm thinking that H=<9> = {1,7,9} (viewed as a cyclic subgroup of Z30) is this correct?
And could someone explain what (b) is asking in other terms?

 

[gerben]

I'm thinking that H=<9> = {1,7,9} (viewed as a cyclic subgroup of Z30) is this correct?
And could someone explain what (b) is asking in other terms?

No that is not correct. Z30 is an additive group so H=<9> contains 9, 9+9, 9+9+9 etc.

Part (b) asks which elements of H could produce H by just adding it to itself repeatedly

 

[tasha10]

so {9,18,27}?

 

[gerben]

No, quite a few more. What do you get if you add 9 to 27 in Z30?