Definition of a generator of a cyclic group?

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October 10th 2011, 06:50 PM
tangibleLime

Definition of a generator of a cyclic group?

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

$<a> = \left \{ a^n | n \in \mathbb{Z} \right \}$

Almost immediately after, it gives an example with $Z_{18}$ , and the generator <2>. but it says...

$<2> = \left\{0, 2, 4, 6, 8, 10, 12, 14, 16\right\}$

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

$<2> = \left \{ 2^n | n \in \mathbb{Z} \right \} = \left \{ \right 2^0, 2^1, 2^2, 2^3, 2^4, 2^5, 2^6 \}= \left \{1,2,4,8,16,14,10 \right \}$

And obviously,
$\left\{0, 2, 4, 6, 8, 10, 12, 14, 16\right\} \neq \left \{1,2,4,8,16,14,10 \right \}$

What?
October 10th 2011, 06:59 PM
alexmahone

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:

$<a> = \left \{ a^n | n \in \mathbb{Z} \right \}$

Almost immediately after, it gives an example with $Z_{18}$ , and the generator <2>. but it says...

$<2> = \left\{0, 2, 4, 6, 8, 10, 12, 14, 16\right\}$

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

$<2> = \left \{ 2^n | n \in \mathbb{Z} \right \} = \left \{ \right 2^0, 2^1, 2^2, 2^3, 2^4, 2^5, 2^6 \}= \left \{1,2,4,8,16,14,10 \right \}$

And obviously,
$\left\{0, 2, 4, 6, 8, 10, 12, 14, 16\right\} \neq \left \{1,2,4,8,16,14,10 \right \}$

What?

In $Z_{18}$ , the group operation is addition, not multiplication. That said, 2 is not a generator of $Z_{18}$ because $gcd(2,\ 18)\neq 1$ .
October 10th 2011, 07:03 PM
tangibleLime

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 $\mathbb{Z}$ , 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

$<a> = \left \{ a^n | n \in \mathbb{Z} \right \}$

to

$<a> = \left \{ n \cdot a | n \in \mathbb{Z} \right \}$ ?
October 10th 2011, 07:05 PM
alexmahone

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 $\mathbb{Z}$ , 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

$<a> = \left \{ a^n | n \in \mathbb{Z} \right \}$

to

$<a> = \left \{ n \cdot a | n \in \mathbb{Z} \right \}$ ?

Yes.
October 10th 2011, 07:16 PM
Deveno

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 $\mathbb{Z}$ , 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 $\mathbb{Z}_n$ , it is usually obvious from context that addition modulo n is intended, since $0 \in \mathbb{Z}_n$ has no multiplicative inverse. this applies to the other sets you mentioned as well.
October 10th 2011, 07:18 PM
tangibleLime

Re: Definition of a generator of a cyclic group?

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