The classification of finite simple groups has been called one of the great intellectual achievements of humanity, but I don't even know one single application of it. Even worse, I know a lot of applications of simple modules over some ring/algebra $A$, but I can barely know an application of them for finite simple groups. When studying modules, one has, for example,
- If $S$ and $T$ are distinct simple modules, then $\operatorname{Hom}(S,T) = 0$, and one can enhance this using Jordan-Holder to prove that, if $M$ and $N$ are modules whose Jordan-Holder decomposition don't have common factors, then $\operatorname{Hom}(M,N)=0$. We may use this, for example, to try to compute some cohomology, also;
- The simple modules form a basis of the $K_0$ group, and therefore if we're interested in, for example, the multiplicative structure of $K_0$ it's enough to compute the (tensor) product of simple modules;
- If the algebra $A$ is basic (i.e. every simple representation is $1$-dimensional), which happens for path algebras, then simple modules have a group structure with respect to the tensor product (so they are an analogue for the Picard group).
For finite simple groups, the only application I know is for the (non)-solubility of polynomials, and it's a quite particular example which uses only $S_n$ and $A_n$. So I have two questions:
- What are some (concrete) applications of (finite simple groups + Jordan-Holder) for general finite groups?
- What are some (concrete) applications of the classifications of finite simple groups?