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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?

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  • $\begingroup$ 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. $\endgroup$ – Parcly Taxel May 13 '17 at 0:20
  • $\begingroup$ @ParclyTaxel When you say broken into simple groups, how do you mean? Not every group has a composition series. $\endgroup$ – Aaron Zolotor May 13 '17 at 0:23
  • $\begingroup$ Every finite group has a composition series. $\endgroup$ – Parcly Taxel May 13 '17 at 0:24
  • $\begingroup$ So not all groups but all finite groups then? $\endgroup$ – Aaron Zolotor May 13 '17 at 0:25
  • $\begingroup$ Yes. The enormous theorem never says anything about infinite groups. $\endgroup$ – Parcly Taxel May 13 '17 at 0:26

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