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Concept
Théorème de Schur-Zassenhaus
Résumé
The Schur–Zassenhaus theorem is a theorem in group theory which states that if is a finite group, and is a normal subgroup whose order is coprime to the order of the quotient group , then is a semidirect product (or split extension) of and . An alternative statement of the theorem is that any normal Hall subgroup of a finite group has a complement in . Moreover if either or is solvable then the Schur–Zassenhaus theorem also states that all complements of in are conjugate. The assumption that either or is solvable can be dropped as it is always satisfied, but all known proofs of this require the use of the much harder Feit–Thompson theorem.
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?" The other part, which is where the composition factors do not have coprime orders, is tackled in extension theory.
The Schur–Zassenhaus theorem was introduced by . Theorem 25, which he credits to Issai Schur, proves the existence of a complement, and theorem 27 proves that all complements are conjugate under the assumption that or is solvable. It is not easy to find an explicit statement of the existence of a complement in Schur's published works, though the results of on the Schur multiplier imply the existence of a complement in the special case when the normal subgroup is in the center. Zassenhaus pointed out that the Schur–Zassenhaus theorem for non-solvable groups would follow if all groups of odd order are solvable, which was later proved by Feit and Thompson. Ernst Witt showed that it would also follow from the Schreier conjecture (see for Witt's unpublished 1937 note about this), but the Schreier conjecture has only been proved using the classification of finite simple groups, which is far harder than the Feit–Thompson theorem.
If we do not impose the coprime condition, the theorem is not true: consider for example the cyclic group and its normal subgroup .
Source officielle
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