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I've just studied the Schur-Zassenhaus theorem (here there is the statement), but I don't understand its importance in the theory of finite groups. Wikipedia for example says:

The Schur–Zassenhaus theorem at least partially answers the question: "In a composition series, how can we classify groups with a certain set of composition factors?"

but for me it's not clear what does it means.

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Do you know what a composition series and composition factor is? – Thomas Andrews Jan 16 '13 at 14:54
Yes I know what they are – Dubious Jan 16 '13 at 15:39

If you know the composition factors consist of the sequence of simple groups $H_1,...,H_n$ with the orders of the groups relatively prime, then you know that the original group must be some chained a chained semi-direct product of those simple groups.

It's certainly not extremely helpful, since the composition factors are simple groups, and the non-commutative simple groups are all of even order. so for this condition to hold, at most one of the $H_i$ can be non-commutative.

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