1. Elements of order

Hi guys, if i say 2 is a primitive root in mod 19, how can i find all the elements of order 6 in Z*19, without finding the order of each element? Thank you for the help

2. Re: Elements of order

Hi Audrey,
If you know a little about groups, it's not too hard. But some computation is necessary, I think. See the attachment:

3. Re: Elements of order

thank you john, i get to know where to start!

4. Re: Elements of order

In general, if we have a primitive root in $(\mathbb{Z}_p)^{\times}$, we know this group is cyclic, and cyclic groups have some special properties.

In particular, we know that if $G = \langle a \rangle$, the order of $a^k$ is $\frac{|G|}{\text{gcd}(k,|G|)}$.

In this example, we have |G| = 18, so we seek k such that 18/(gcd(k,18)) = 6, so that gcd(k,18) = 3. How many such k are there?

Well, clearly k = 3 works, and of course, we need only check the following: 6,9,12,and 15. We can eliminate the even numbers straight-away, leaving just 9 and 15. Since 9|18, that leaves just 15, and indeed gcd(15,18) = 3.

So we have two such k: 3 and 15. So the elements of order 6 are precisely: 23 = 8, and 215 = 85 = (82)(82)(8) = (7)(7)(8) = (7)(18) = 7(-1) = -7 = 12.