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,


1. 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;
2. 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;
3. 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:


1. What are some (concrete) applications of (finite simple groups + Jordan-Holder) for general finite groups?
2. What are some (concrete) applications of the classifications of finite simple groups?



Why are finite simple groups useful?
There's an entire book on this subject, "Applying the Classification of Finite Simple Groups: A User’s Guide" by Stephen D. Smith, published through the AMS, though you can find a draft version [here](http://homepages.math.uic.edu/%7Esmiths/book.pdf).


The applications are not as simple as they are for modules, but many questions can be settled by invoking the classification of finite simple groups. For example, you can invoke the classification to in turn classifying 2-transitive groups. The only known proof of the Schreier conjecture relies on the classification. The book has many more applications.


