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

Question

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.

Answer 1

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.