# Thread: List elements of order 10

1. ## List elements of order 10

List all elements of $\displaystyle Z_{40}$ with order 10.

My solution: By a theorem, I know that the unique subgroup of order 10 is <40/10> = <4> = {0,4,8,12,16,20,24,28,32,36>

List all elements of $\displaystyle Z_{40}$ with order 10.

My solution: By a theorem, I know that the unique subgroup of order 10 is <40/10> = <4> = {0,4,8,12,16,20,24,28,32,36>

Think about it. It has to have ten members and it has to be closed and it has to include 0. That's going to be awfully hard to do with a different subset than the one you listed.

That's the only one.

-Dan

List all elements of $\displaystyle Z_{40}$ with order 10.

My solution: By a theorem, I know that the unique subgroup of order 10 is <40/10> = <4> = {0,4,8,12,16,20,24,28,32,36>

Suppose $\displaystyle k$ is an element of $\displaystyle Z_{40}$ of order $\displaystyle 10$, then:

$\displaystyle 10k \equiv 0 \mod 40$

or there exists a $\displaystyle \lambda \in \bold{N}$ such that:

$\displaystyle 10k=\lambda 40$

which imples that $\displaystyle k$ is a multiple of $\displaystyle 4$.

But not all multiples of $\displaystyle 4$ are of order $\displaystyle 10$, for example $\displaystyle 8$ is of order $\displaystyle 5$.

RonL

4. Given $\displaystyle G=\mathbb{Z}_{40}$ if $\displaystyle a\in G$ then $\displaystyle \mbox{ord}(a) = \frac{40}{\gcd(a,40)}$. You can take it from here.

5. So does that means the elements are {4,12,28,36}?

But I use a theorem in the book, in which says ord(a^k) = n/gcd(k,n).

Then I get k = 1, 3, 7, 9.

So shouldn't the answers be {4, 4^3, 4^7, 4^9}?

So does that means the elements are {4,12,28,36}?

But I use a theorem in the book, in which says ord(a^k) = n/gcd(k,n).

Then I get k = 1, 3, 7, 9.

So shouldn't the answers be {4, 4^3, 4^7, 4^9}?
So what I posted (theorem) is basically the same.

So we have $\displaystyle \mbox{ord}(a) = \frac{40}{\gcd(a,40)}$.

Since we have the order 10. Thus, $\displaystyle \gcd(a,40)=10$. Now find all such so that this is 10.