Puig subgroup


In finite group theory, a branch of mathematics, 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 LG 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
They have the property that
  • L0 ⊆ L2 ⊆ L4... ⊆...L5 ⊆ L3 ⊆ L1
The Puig subgroup L is the intersection of the subgroups Ln for n odd, and the subgroup L* is the union of the subgroups Ln for n even.

Properties

Puig proved that if G is a 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 of the Puig subgroup of S is a normal subgroup of G.