# Building Groups out of Simple Groups?

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!

Edit

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?

• We know that all groups can be broken into simple groups. The real innovation of the 20th century was a complete characterisation of what those finite simple groups were - the enormous theorem. – Parcly Taxel May 13 '17 at 0:20
• @ParclyTaxel When you say broken into simple groups, how do you mean? Not every group has a composition series. – Aaron Zolotor May 13 '17 at 0:23
• Every finite group has a composition series. – Parcly Taxel May 13 '17 at 0:24
• So not all groups but all finite groups then? – Aaron Zolotor May 13 '17 at 0:25
• Yes. The enormous theorem never says anything about infinite groups. – Parcly Taxel May 13 '17 at 0:26