
Puig subgroup

In mathematical finite group theory, the Puig subgroup, introduced by , 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
They have the property that
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".