Sylow theorems are used to prove groups of specific orders have particular structures or are simple. For example, no simple group of order 12 exists, and every group of order p² is abelian.
The Sylow theorems — which you have already studied — give you three tools: existence of Sylow p-subgroups, their conjugacy, and a congruence constraint on their count n_p. Applications means weaponizing these tools to force structural conclusions about groups whose orders factor in particular ways. The general strategy is: compute what n_p *must* be, use the constraints to show it must equal 1, and conclude that the unique Sylow subgroup is normal — giving you a normal subgroup to work with.
Consider a group G of order 12 = 2² × 3. The number of Sylow 3-subgroups satisfies n_3 ≡ 1 (mod 3) and n_3 | 4, so n_3 ∈ {1, 4}. The number of Sylow 2-subgroups satisfies n_2 ≡ 1 (mod 2) and n_2 | 3, so n_2 ∈ {1, 3}. If n_3 = 4, those four subgroups each have order 3 and pairwise trivial intersection, contributing 4 × 2 = 8 non-identity elements of order 3. That forces n_2 = 1 (only 4 elements remain). So in either case — n_3 = 1 or n_2 = 1 — there is a normal Sylow subgroup. No group of order 12 can be simple (have no nontrivial normal subgroups), because we always find one.
For groups of order p², the argument is different. Every group of order p² is abelian. The proof uses the fact that the center Z(G) is nontrivial (a standard result from the class equation), so |Z(G)| is p or p². If |Z(G)| = p², then G = Z(G) is abelian. If |Z(G)| = p, then G/Z(G) has order p and is therefore cyclic, but a group G is abelian whenever G/Z(G) is cyclic — a contradiction that forces |Z(G)| = p² after all.
The meta-skill to internalize is counting with Sylow constraints to force normality. Many "show this group is not simple" or "classify groups of order n" proofs follow the same skeleton: (1) identify the possible Sylow subgroups by applying the divisibility and congruence constraints, (2) count elements to show the options are incompatible unless one n_p = 1, (3) conclude the group has a proper normal subgroup and therefore cannot be simple, or (4) use the normal subgroup's structure to classify the full group as a direct or semidirect product. Mastery is recognizing which prime p to target first for the tightest constraint.