If one read's Gallian's Abstract Algebra book then they would find that the chapter for Sylow Theorem's is quite hyped up. However, I am unable to understand the big picture of why Sylow subgroups and sylow theorems are important in group theory as a whole. Could some explain the big picture in simple terms?
Why are Sylow Theorems and Sylow subgroups significant
group-theorysoft-questionsylow-theory
Best Answer
For a tourist to a new country it is important to know what are the scenic places are. When studying groups one wants to know what its subgroups are.
Lagrange's theorem is the first theorem in this regard: for a subset to be a subgroup its cardinality should divide the cardinality(order) of the group. This is applicable to ALL groups without exception.
On the other hand when taking a divisor of the order of the group, like 6 for the alternating group $A_4$ there is no subgroup of that order.
So it was considered important to know which divisors of the order of the group, guarantee existence of subgroups of that order. Sylow theorem is one such theorem in that direction, and is applicable to ALL groups.
For special kind of groups such as abelian groups it is always possible to find subgroups of any desired order dividing the order of the full group.
The beauty of Sylow theorem is that it also says if we find more than one subgroup of that order they will be conjugate subgroups.
This means, in case in a group of order $810$ we find two subgroups of order $81$, with one of them cyclic, the other can't be direct product of two cyclic groups of order $9$.