group containing normal subgroups of orders $3$ and $5$ contains element of order $15$

Question

I want to prove that 'Let $G$ be a group containing normal subgroups of orders $3$ and $5$, respectively. Prove $G$ contains an element of order $15$'.

How can I use this fact for proof: 'Let $r$ and $s$ be relatively prime integers. A cyclic group of order $rs$ is isomorphic to the product of a cyclic group of order $r$ and a cyclic group of order $s$.' ?

I first thought that I get the proof obviously, but then I saw 'normal' in question, and stopped.

Also, the cyclic subgroup generated by that order $15$ element has to be normal in $G$?

Answer 1

Here is a roadmap:

• Prove that $H_3 H_5 = H_5 H_3$

• Prove that $H_3 H_5$ is a normal subgroup

• Prove that $H_3 H_5 \cong H_3 \times H_5 \cong C_3 \times C_5 \cong C_{15}$