Puig subgroup

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title: "Puig subgroup" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["finite-groups"] description: "Characteristic subgroup in mathematical finite group theory" topic_path: "general/finite-groups" source: "https://en.wikipedia.org/wiki/Puig_subgroup" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Characteristic subgroup in mathematical finite group theory ::

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 L**G(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
• 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 **-core of G is trivial, then the center Z(L(S)) of the Puig subgroup of S is a normal subgroup of G.

References

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::