This is only mostly a conceptual question.
When beginning to study simple groups you will repeatedly hear something along the lines "we use simple groups to build up larger more complicated groups". I want to know exactly what we mean by this? (With examples if possible)
A la Jordan-Holder we know that every finite simple group has a composition series (whose factors are simple groups) and, in general, that if a group has a composition series it is of constant length and the factors have, at most, been permuted. However, this doesn't even provide a way of identifying a group as I imagine many groups can have the same composition factors?
Looking to really develop my intuition on this one. Cheers!
Say $G$ is a finite group. Then $G$ has a composition series. Is $G$ always isomorphic to the direct product (sum) of its composition factors?