How many elements in a group Z*81?

• Oct 25th 2008, 07:08 PM
apsis
How many elements in a group Z*81?
The subject says it all. The question is How many elements does the group Z*81 have?

I'm really not too sure what the question is asking. I know if I laid the table out it'd be 81 x 81, but there would only be the integers from 0 to 80. Does that mean the answer to the question is 80? Or am I missing something here?

Thanks!
• Oct 25th 2008, 07:10 PM
Jhevon
Quote:

Originally Posted by apsis
The subject says it all. The question is How many elements does the group Z*81 have?

I'm really not too sure what the question is asking. I know if I laid the table out it'd be 81 x 81, but there would only be the integers from 0 to 80. Does that mean the answer to the question is 80? Or am I missing something here?

Thanks!

you mean $\displaystyle \mathbb{Z}_{81}^\times$? it has $\displaystyle \phi (81)$ elements
• Oct 25th 2008, 07:18 PM
ThePerfectHacker
Quote:

Originally Posted by apsis
The subject says it all. The question is How many elements does the group Z*81 have?

I'm really not too sure what the question is asking. I know if I laid the table out it'd be 81 x 81, but there would only be the integers from 0 to 80. Does that mean the answer to the question is 80? Or am I missing something here?

Thanks!

A number $\displaystyle 1\leq a \leq 81$ is invertible iff there is $\displaystyle x$ so that $\displaystyle ax\equiv 1(\bmod 81)$. In order to be able to solve this congruence we require that $\displaystyle \gcd(a,81)=1$. Thus, we are asking how many elements are there between 1 and 81 that are relatively prime to 81. And this, by definition, is $\displaystyle \phi (81)$. Now using the formula $\displaystyle \phi (p^n) = p^n - p^{n-1}$ where $\displaystyle p$ is a prime, we find that $\displaystyle \phi (81) = \phi (3^4) = 81 - 27 = 54$
• Oct 25th 2008, 07:22 PM
apsis
Sorry we usually use $\displaystyle \mathbb{Z}_{81}^*$ to mean $\displaystyle \mathbb{Z}_{81}^\times$ so yes that is what I mean.

And so your basically saying that the size of the group is the number of integers less than 81 that are coprime to 81?

Could you explain a little? I'm not really sure what is meant by the size of the group. Like the number of elements that are in the group?

So if you take all the elements from 0 to 80. And then break them into co-primes, then that gives you the number of elements since other elements are really just results of the combination of those primes?

Thanks!

Wasn't fast enough in my response. Thanks guys you rock!
• Oct 25th 2008, 07:29 PM
Jhevon
Quote:

Originally Posted by apsis
Sorry we usually use $\displaystyle \mathbb{Z}_{81}^*$ to mean $\displaystyle \mathbb{Z}_{81}^\times$ so yes that is what I mean.

okie dokie

Quote:

And so your basically saying that the size of the group is the number of integers less than 81 that are coprime to 81?
yes, the (multiplicative) group $\displaystyle \mathbb{Z}_n^\times$ denotes the group with the elements of $\displaystyle \mathbb{Z}_n$ that are relatively prime to $\displaystyle n$

Quote:

Could you explain a little? I'm not really sure what is meant by the size of the group. Like the number of elements that are in the group?
yes, "size" and "order" of a group refers to the number of elements in the group. at least in the finite case.

Quote:

So if you take all the elements from 0 to 80. And then break them into co-primes, then that gives you the number of elements since other elements are really just results of the combination of those primes?

Thanks!