Editing Sylow theorems (section)

== Algorithms ==
The problem of finding a Sylow subgroup of a given group is an important problem in [[computational group theory]].

One proof of the existence of Sylow ''p''-subgroups is constructive: if ''H'' is a ''p''-subgroup of ''G'' and the index [''G'':''H''] is divisible by ''p'', then the normalizer ''N'' = ''N<sub>G</sub>''(''H'') of ''H'' in ''G'' is also such that [''N'' : ''H''] is divisible by ''p''.  In other words, a polycyclic generating system of a Sylow ''p''-subgroup can be found by starting from any ''p''-subgroup ''H'' (including the identity) and taking elements of ''p''-power order contained in the normalizer of ''H'' but not in ''H'' itself. The algorithmic version of this (and many improvements) is described in textbook form in Butler,{{sfn|Butler|1991|loc=Chapter 16}} including the algorithm described in Cannon.{{sfn|Cannon|1971}} These versions are still used in the [[GAP computer algebra system]].

In [[permutation group]]s, it has been proven, in Kantor{{sfn|Kantor|1985a}}{{sfn|Kantor|1985b}}{{sfn|Kantor|1990}} and Kantor and Taylor,{{sfn|Kantor|Taylor|1988}} that a Sylow ''p''-subgroup and its normalizer can be found in [[polynomial time]] of the input (the degree of the group times the number of generators).  These algorithms are described in textbook form in Seress,{{sfn|Seress|2003}} and are now becoming practical as the constructive recognition of finite simple groups becomes a reality.  In particular, versions of this algorithm are used in the [[Magma computer algebra system]].