1. ## Sylow p-group

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

2. If $\displaystyle H,K$ are normal subgroups then $\displaystyle HK$ is a normal subgroup. Also if $\displaystyle H\cap K = \{ e \}$ then $\displaystyle HK \simeq H\times K$.
Thus, if $\displaystyle \gcd (|H|,|K|) = 1$ then $\displaystyle H\cap K = \{ e \}$. This result generalizes to finitely many groups.
In your case you have the Sylow subgroups $\displaystyle G_{p^a},G_{q^b}, ...$ all of them are normal because there is only one, and furthermore, all of them have trivial group intersection because their orders are relatively prime.