- #1

- 78

- 0

Also, Consider Zn = {0,1,....,n-1}

a. show that an element k is a generator of Zn if and only if k and n are relatively prime.

b. Is every subgroup of Zn cyclic? If so, give a proof. If not, provide an example.

- Thread starter halvizo1031
- Start date

- #1

- 78

- 0

Also, Consider Zn = {0,1,....,n-1}

a. show that an element k is a generator of Zn if and only if k and n are relatively prime.

b. Is every subgroup of Zn cyclic? If so, give a proof. If not, provide an example.

- #2

CompuChip

Science Advisor

Homework Helper

- 4,302

- 47

What possible ways to prove commutativity do you know of?

For the second one, I suggest starting with the "<==" implication (i.e. assume that k and n are relatively prime and show that k generates

- #3

- 78

- 0

What possible ways to prove commutativity do you know of?

For the second one, I suggest starting with the "<==" implication (i.e. assume that k and n are relatively prime and show that k generatesZ_{n}.

for the first one, we can show commutativity with a multiplication table. How else?

for the second one, i want to start with ==> and say that the order of k is n/(m,n). but how can i show it?

- #4

CompuChip

Science Advisor

Homework Helper

- 4,302

- 47

The definition of commutativity is that xy = yx for any two elements x and y.

Can you explicitly show this in the case given?

- Last Post

- Replies
- 6

- Views
- 2K

- Last Post

- Replies
- 4

- Views
- 2K

- Last Post

- Replies
- 3

- Views
- 9K

- Last Post

- Replies
- 3

- Views
- 2K

- Last Post

- Replies
- 1

- Views
- 1K

- Last Post

- Replies
- 2

- Views
- 2K

- Last Post

- Replies
- 3

- Views
- 2K

- Last Post

- Replies
- 4

- Views
- 1K

- Last Post

- Replies
- 1

- Views
- 1K

- Last Post

- Replies
- 3

- Views
- 1K