Definition of a generator of a cyclic group?

My book defines a generator *a* of a cyclic group as:

Almost immediately after, it gives an example with , and the generator <2>. but it says...

Where are those numbers (0, 2, 4, ..., 16) coming from? If I follow the definition of a generator exactly,

And obviously,

What?

Re: Definition of a generator of a cyclic group?

Quote:

Originally Posted by

**tangibleLime** My book defines a generator

*a* of a cyclic group as:

Almost immediately after, it gives an example with

, and the generator <2>. but it says...

Where are those numbers (0, 2, 4, ..., 16) coming from? If I follow the definition of a generator exactly,

And obviously,

What?

In , the group operation is addition, not multiplication. That said, 2 is not a generator of because .

Re: Definition of a generator of a cyclic group?

Grr, I keep typing responses and the forum erases them when I hit reply...??

Anyways,

So how do I know it's addition? Is it just standard and assumed that with , addition is always used for cyclic groups? What about Q, C, R, etc?

So when it's addition, the definition of a generator changes from

to

?

Re: Definition of a generator of a cyclic group?

Quote:

Originally Posted by

**tangibleLime** So how do I know it's addition? Is it just standard and assumed that with

, addition is always used for cyclic groups? What about Q, C, R, etc?

Yes, it is addition unless specified otherwise.

Quote:

So when it's addition, the definition of a generator changes from

to

?

Yes.

Re: Definition of a generator of a cyclic group?

Quote:

Grr, I keep typing responses and the forum erases them when I hit reply...??

Anyways,

So how do I know it's addition? Is it just standard and assumed that with

, addition is always used for cyclic groups? What about Q, C, R, etc?

as other have pointed out in other threads, a set is not a group, so a multiplication must be specified.

however, with , it is usually obvious from context that addition modulo n is intended, since has no multiplicative inverse. this applies to the other sets you mentioned as well.

Re: Definition of a generator of a cyclic group?

Ah, now that I think about it a bit more, that does make sense. Thanks!