Puig subgroup
{{Short description|Characteristic subgroup in mathematical finite group theory}}
In finite group theory, a branch of mathematics, the Puig subgroup, introduced by {{harvs|txt|last=Puig|year=1976}}, is a characteristic subgroup of a p-group analogous to the Thompson subgroup.
Definition
If H is a subgroup of a group G, then LG(H) is the subgroup of G generated by the abelian subgroups normalized by H.
The subgroups Ln of G are defined recursively by
- L0 is the trivial subgroup
- Ln+1 = LG(Ln)
They have the property that
- L0 ⊆ L2 ⊆ L4... ⊆ ...L5 ⊆ L3 ⊆ L1
The Puig subgroup L(G) is the intersection of the subgroups Ln for n odd, and the subgroup L*(G) is the union of the subgroups Ln 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 {{prime|p}}-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=Bender | first1=Helmut | last2=Glauberman | first2=George | author2-link=George Glauberman | title=Local analysis for the odd order theorem | publisher=Cambridge University Press | series=London Mathematical Society Lecture Note Series | isbn=978-0-521-45716-3 |mr=1311244 | year=1994 | volume=188|chapter=Appendix B - The Puig Subgroup|pages= 139–144|url=https://books.google.com/books?id=DvTw3KMSLdsC&pg=PA139}}
- {{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}}