Puig subgroup

In mathematical finite group theory, the Puig subgroup, introduced by Puig (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 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

L₀ is the trivial subgroup
L_n+1 = L_G(L_n)

They have the property that

L₀ ⊆ L₂ ⊆ L₄... ⊆ ...L₅ ⊆ L₃ ⊆ L₁

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 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

Bender, Helmut; Glauberman, George (1994), "Appendix B - The Puig Subgroup", Local analysis for the odd order theorem, London Mathematical Society Lecture Note Series, vol. 188, Cambridge University Press, pp. 139–144, ISBN 978-0-521-45716-3, MR 1311244
Puig, Luis (1976), "Structure locale dans les groupes finis", Bulletin de la Société Mathématique de France (47): 132, ISSN 0037-9484, MR 0450410