If $|G|=p^rm$ with $(p,m)=1$, suppose that $x\in G$ is an element such that $o(x)=p^{r_1}m_1$ with $r_1>0$ and $(m_1,p)=1$. I dont understand why exist $a,b\in G$ such that:
1) $a$ has order a power of $p$
2) $b$ has order coprime with $p$
3) $x=ab$ and $[a,b]=1$
This fact is often used in proofs and it is presented as an obviuosly fact.