Thread: Cyclic groups modulo

Hi all,

Brief intro to what im talking about....

If n 2 is an integer, then Zn* denotes the set of invertible elements in the ring Zn. That is, it denotes the numbers in {1,2....n-1} which are coprime to n. The set Zn* is a group under multiplication modulo n.

Does anyone know how to determine whether or not these groups are cyclic?: Z8*, Z9*, Z10*, Z12*

Maybe someone can walk me through the first couple and i can try the others for myself!

Thanks!

Originally Posted by sirellwood

Hi all,

Brief intro to what im talking about....

If n 2 is an integer, then Zn* denotes the set of invertible elements in the ring Zn. That is, it denotes the numbers in {1,2....n-1} which are coprime to n. The set Zn* is a group under multiplication modulo n.

Does anyone know how to determine whether or not these groups are cyclic?: Z8*, Z9*, Z10*, Z12*

Maybe someone can walk me through the first couple and i can try the others for myself!

Thanks!

This is standard stuff you can find in any decent book in group theory or general algebra. Read the folowing, too:
Multiplicative group of integers modulo n - Wikipedia, the free encyclopedia

Tonio

here is how you would proceed for U(Z8): the elements co-prime to 8 are the odd integers {1,3,5,7} (since 8 is a power of 2). this is a group of order 4, for it to be cyclic, you would need an element of order 4. it suffices to check 3,5 and 7. 3^2 = 1, 5^2 = 1, and 7^2 = 1, so U(Z8) is not cyclic.

U(Z9) is even easier: it has 6 elements {1,2,4,5,7,8}. we know that the multiplication is commutative, and any abelian group of order 6 is cyclic.