# Non cyclic group of order 125 which has an element of order 25.

I am trying to find a non cyclic group of order 125 which has an element of order 25.

I know product of cyclic groups can be used, but I am not sure whether an element of order 25 exists.

is there any generalized method to come up with examples for any given group with any given order? like a non cyclic group of order 63, with element of order 21.

• Not sure if this would work, but maybe try a semi-direct product? Commented Mar 6, 2017 at 9:04
• What about $C_{25} \times C_5$ and $C_{21} \times C_3$? Consider the element $(1,0)$. Commented Mar 6, 2017 at 9:04
• @SquirtleSquad Yeah you're right. Commented Mar 6, 2017 at 9:07

Group: $\mathbb{Z}_{25}\times\mathbb{Z}_{5}$
Order $25$ element: $(1,0)$
In general for non-cyclic group of order $pq$ with $p,q$ not coprime you can take $\mathbb{Z}_{p}\times\mathbb{Z}_{q}$ then $(1,0)$ will be of order $p$ as required.
• yes, $(0,1)$, you could also just swap $p,q$ Commented Mar 6, 2017 at 9:19