# Cyclic groups?

• December 26th 2007, 10:04 AM
Jason Bourne
Cyclic groups?
(a) Define what it means for s group to be cyclic.

(b) Give an example of an infinite cyclic group

(c) Determine whether the group U[15] = {1,2,4,7,8,11,13,14}, where multiplication is mod 15, is cyclic.
• December 26th 2007, 10:11 AM
ThePerfectHacker
Quote:

Originally Posted by Jason Bourne
(a) Define what it means for s group to be cyclic.

That is something you need to know from your textbook.

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(b) Give an example of an infinite cyclic group
$\mathbb{Z}$
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(c) Determine whether the group U[15] = {1,2,4,7,8,11,13,14}, where multiplication is mod 15, is cyclic.
Can you find a generator?
• December 27th 2007, 12:42 PM
Jason Bourne
No, I cannot find any generators, so the group is not cyclic. (Rofl)
• December 27th 2007, 01:05 PM
ThePerfectHacker
Quote:

Originally Posted by Jason Bourne
No, I cannot find any generators, so the group is not cyclic. (Rofl)

That is correct. In general if $n>1$ then the group $\mathbb{Z}_n^{\text{x}}$ is cyclic if and only if $n=2,4,p^k,2p^k$ where $p$ is an odd prime.