# Thread: The integers modulo m with zero removed are a cyclic group for multiplication modulo

1. ## The integers modulo m with zero removed are a cyclic group for multiplication modulo

Hi: This is An Introduction to the Theory of Groups by George Polites, example 11, page 9: please see first attachment below. And now he gives this exercise: please see second attachment below.

I say any prime will do. If m is a prime then G will be a field for
sum and product modulo m. In a field the elements different from zero are
a group for multiplication and this group is cyclic. I remember it was not a trivial matter to
prove that group is cyclic. How can the author put such a difficult
problem at such an elementary stage? He has just shown Lagrange theorem!

The fact is I cannot solve the exercise. I chose this book because I hoped
it would be easy but maybe I am wrong.

2. ## Re: The integers modulo m with zero removed are a cyclic group for multiplication mod

If m is not prime then we can factor m= pq where p and q are also integers. Both p and q are less than m of course so in $\displaystyle Z^m$. What is pq (mod m)? What does that tell you about the multiplicative inverses of p and q in this group? Do you see that this cannot happen if m is prime?

3. ## Re: The integers modulo m with zero removed are a cyclic group for multiplication mod

Is $Z^m$ the multiplicative group ${1, 2, 3,...,m-1}$?

4. ## Re: The integers modulo m with zero removed are a cyclic group for multiplication mod

pq= bm + r, $0<= r < m$. pq is r. Or better said, pq = 0 mod m (not pq = 1) in which case $p^{-1} = q$. If m is prime I cannot factor. I do not quite understand.

EDIT: the exersize statement is misleading. I should find conditions on m to make G a group. The author says a CYCLIC group. If it is a group of course it will be cyclic. Every i in G should be prime relative to m. In this way i will have an inverse in G. To achieve this, m must be prime. That's all.

5. ## Re: The integers modulo m with zero removed are a cyclic group for multiplication mod

Originally Posted by STF92
Is $Z^m$ the multiplicative group ${1, 2, 3,...,m-1}$?
That is only a group if m is prime.

6. ## Re: The integers modulo m with zero removed are a cyclic group for multiplication mod

I agree ("I say any prime will do.", post #1). How do I mark the thread solved?