For a finite group $G$ of size $|G| = p_1^{i_1} \times\ldots\times p_n^{i_n}$, where $p_i$s are distinct primes, can we use $G$-action on Sylow $p_i$-subgroups to define a faithful action of $G$? Under what conditions on $G$ this is possible? What if we do not limit ourselves to action by conjugation and also consider coset action?
The following post seems relevant when the action is limited to a single-family of $p_i$, but does not consider simultaneous action on multiple set of Sylow subgroups.
Group action on Sylow subgroups
Best Answer
If you act by conjugacy, the elements of the centre $Z(G)$ of $G$ will fix every $p_{i}$-Sylow subgroup, for each $i$, so if $Z(G) \ne 1$ the action is not faithful.
On the other hand, if you act by multiplication on the coset of a $p_{i}$-Sylow subgroup $P$, for some $i$, the kernel of the action will be the core $\bigcap_{g \in G} P^{g}$ of $P$. So this is a $p_{i}$-subgroup. If the order of the group $G$ is divisible by at least two distinct primes, then, you have a faitfhul action here.