Could anyone help me on this question:

Let $\displaystyle G$ be finite group of order $\displaystyle |G| = p^a q^b ... t^d $ where $\displaystyle p,q, ... t$ are distinct primes and $\displaystyle a,b,.. d$ are positive integers. Suppose that $\displaystyle G$ has only one Sylow p-group $\displaystyle G_{p^a} $for each prime $\displaystyle p$ that divides $\displaystyle |G|$.

Show that $\displaystyle G = G_{p^a} \times G_{q^b} \times ... \times G_{t^d}$

Is it true that $\displaystyle gcd(p^a, q^b)=1$ , for any distinct primes?

If it is true, can I use Sylow theorem and because G has only 1 Sylow p-group for each prime? Therefore

$\displaystyle G = G_{p^a} \times G_{q^b} \times ... \times G_{t^d}$

Thank you