In mathematical finite group theory, the Puig subgroup, introduced by , is a characteristic subgroup of a ''p''-group analogous to the

Thompson subgroup In mathematical finite group theory, the Thompson subgroup J(P) of a finite ''p''-group ''P'' refers to one of several characteristic subgroups of ''P''. originally defined J(P) to be the subgroup generated by the abelian subgroups of ''P'' of m ...

Definition

If ''H'' is a subgroup of a group ''G'', then ''L''_''G''(''H'') is the subgroup of ''G'' generated by the abelian subgroups normalized by ''H''. The subgroups ''L''_''n'' of ''G'' are defined recursively by *''L''₀ is the trivial subgroup *''L''_''n''+1 = ''L''_''G''(''L''_''n'') They have the property that *''L''₀ ⊆ ''L''₂ ⊆ ''L''₄... ⊆ ...''L''₅ ⊆ ''L''₃ ⊆ ''L''₁ The Puig subgroup ''L''(''G'') is the intersection of the subgroups ''L''_''n'' for ''n'' odd, and the subgroup ''L''_*(''G'') is the union of the subgroups ''L''_''n'' for ''n'' even.

Properties

Puig proved that if ''G'' is a (solvable) group of odd order, ''p'' is a prime, and ''S'' is a Sylow ''p''-subgroup of ''G'', and the '-core of ''G'' is trivial, then the center ''Z''(''L''(''S'')) of the Puig subgroup of ''S'' is a normal subgroup of ''G''.

References

* *{{Citation , last1=Puig , first1=Luis , title=Structure locale dans les groupes finis , url=http://www.numdam.org/item?id=MSMF_1976__47__5_0 , mr=0450410 , year=1976 , journal=Bulletin de la Société Mathématique de France , issn=0037-9484 , issue=47 , pages=132 Finite groups