# Thread: How to check group of given order cyclic or not?

1. ## How to check group of given order cyclic or not?

if we've given any order of group then how can we check that is it cyclic or not example how can we check group of order 255 and 465 and 3000 order cyclic or not.Please help

2. Originally Posted by reflection_009
if we've given any order of group then how can we check that is it cyclic or not example how can we check group of order 255 and 465 and 3000 order cyclic or not.Please help
If $\displaystyle |G| = pq$ with $\displaystyle q>p$ primes and $\displaystyle q\not \equiv 1 (\bmod p)$ then $\displaystyle G\simeq \mathbb{Z}_{pq}$. Otherwise $\displaystyle q\equiv 1(\bmod p)$ then it is not true.

Now $\displaystyle 255 = 5\cdot 51$ there the group is not necessarily cyclic.

Note $\displaystyle 465 = 3\cdot 5\cdot 31$ let $\displaystyle H$ be a non-cyclic group of order $\displaystyle 3\cdot 31$. Then $\displaystyle | \mathbb{Z}_5 \times H| = 465$ and $\displaystyle \mathbb{Z}_5 \times H$ is not cyclic.

If $\displaystyle |G| = 2n$ where $\displaystyle n\geq 3$ then we can construct a non-abelian group $\displaystyle D_n$ - the dihedral.
Therefore a group of order $\displaystyle 3000$ is not necessarily cyclic.

3. But 255 is is cyclic i've read in book.