# Thread: Cyclic Groups - Is it a Cyclyc group?

1. ## Cyclic Groups - Is it a Cyclyc group?

$\displaystyle Z_{10}\times Z_{15}$ , while Z is the integers (I believe the action should be '+' for it to be a group, otherwise, for instance in multiplication, (0,0) has no inverse element).

How do I generally check whether it's a cyclic group? It has 150 elements

Thanks

$\displaystyle Z_{10}\times Z_{15}$ , while Z is the integers (I believe the action should be '+' for it to be a group, otherwise, for instance in multiplication, (0,0) has no inverse element).

How do I generally check whether it's a cyclic group? It has 150 elements

Thanks
You need to find a generator. However, this group is NOT a cyclic group.

Assume $\displaystyle (a, b)$ generates your group. Then $\displaystyle (a, b)^n \neq (0, 0)$ for all $\displaystyle 0<n<150$ (why?). However, take $\displaystyle n=30$...

Something to think about: Why does this work? Why the number 30?

3. Nice, gcd(10,15)=30.
And, since (a,b) is the generator, then |(a,b)|=150.

Got it ! Thank you very much!