Otto H.Kegel’sBeloved Contributions to Sylow Theory –Third edition

Let p be aprime.A p-group is agroup all of whose elementshave order a powerof p and a p’-group is a group all of whose elements have order prime to p, that is, not containing any element of order p. A p-group is finite if and only if its order is a power of p. Every group theorist knows what are well-known as the three Sylow theorems: Any finite group G haswhich are called Sylow p-subgroups and are not only maximal with respect to(w.r.t.) orderbut also w.r.t. inclusion, whence every p-subgroup of G is contained in at least oneSylow p-subgroup, and all the Sylow p-subgroups are conjugate, that is, if S1and S2areSylowp-subgroupsofG,thenthereexistsanelement x of G suchthat x-1S1x= S2.* Theset of all Sylow p-subgroups of a group G is denoted by SylpG. The theorems and the maximalp-subgroups arenamed after Ludvig Sylow, thegreat Norwegianmathematicianwho discovered them and published them in December 1872 (see [30.] and https://en.wikipedia.org/wiki/Peter_Ludvig_Sylow). ASylowp-subgroupofanygroupisap-subgroup,whichismaximalw.r.t. inclusion. Every p-subgroup is contained in at least one Sylow p-subgroup. A group satisfies the Sylow Theorem for the Prime p or the Sylow p-Theorem, if all of its Sylowp-subgroupsareconjugate,anditsatisfiestheStrongSylowTheoremforthePrimep,ifeach ofitssubgroupssatisfiestheSylowp-Theorem.Alocallyfinitegroupisagroupallofwhose finitelygeneratedsubgroupsarefinite.SylowTheoryofLocallyFiniteGroupsstudieswhen theysatisfytheSylowp-TheoremandwheneventheStrongSylowp-Theoremanddetermines thestructureofthosegroups.Acentralconcepttothatendisthep-uniquenesssubgroupofa locallyfinitegroup,whichisafinitep-subgroupbeingcontainedinauniqueSylowp-subgroup.