# Cyclic Groups - Is it a Cyclyc group?

• Aug 12th 2010, 02:13 AM
Cyclic Groups - Is it a Cyclyc group?
$Z_{10}\times Z_{15}$ , 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 (Surprised)

Thanks (Happy)
• Aug 12th 2010, 02:51 AM
Swlabr
Quote:

$Z_{10}\times Z_{15}$ , 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 (Surprised)

Thanks (Happy)

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

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

Something to think about: Why does this work? Why the number 30?
• Aug 12th 2010, 03:04 AM
Nice, gcd(10,15)=30.
And, since (a,b) is the generator, then |(a,b)|=150.

Got it :) ! Thank you very much!
• Aug 12th 2010, 03:10 AM
Swlabr
Quote: