tasha10 said: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?
A cyclic subgroup of Z30 is a subset of the integers from 1 to 30 that can be generated by repeatedly adding a single element to itself. In other words, it is a group that contains a single element that can be multiplied by itself a certain number of times to create all the other elements in the group.
There are 30 elements in a cyclic subgroup of Z30, as the name suggests. This is because Z30 is a cyclic group itself, and all cyclic subgroups of a cyclic group have the same number of elements.
The identity element in a cyclic subgroup of Z30 is the number 1. This is because multiplying any other element in the subgroup by 1 will result in that same element, making it the identity element.
The generator of a cyclic subgroup of Z30 is any element that, when multiplied by itself a certain number of times, can generate all the other elements in the subgroup. In this case, any number coprime to 30 (such as 7, 11, or 13) can be a generator of the subgroup.
A cyclic subgroup of Z30 is essentially the same as the integers modulo 30, as both are groups with 30 elements that can be generated by repeatedly adding a single element to itself. However, the difference is that while a cyclic subgroup of Z30 is a subset of the integers, the integers modulo 30 represent the entire group of integers from 1 to 30 under modular arithmetic.