[Paper Review] Unramified Brauer groups and isoclinism

Primož Moravec|arXiv (Cornell University)|Mar 12, 2012

Algebraic Geometry and Number Theory4 references5 citations

TL;DR

This paper proves that isoclinic finite groups have isomorphic Bogomolov multipliers, a key invariant in rationality problems for group actions on fields. Using a combinatorial description of the Bogomolov multiplier via the kernel of the commutator map on the non-abelian exterior square, the authors establish an isomorphism between the corresponding kernels for isoclinic groups, thereby affirming a conjecture on the invariance of this group under isoclinism.

ABSTRACT

We show that if $G_1$ and $G_2$ are isoclinic groups, then their Bogomolov multipliers are isomorphic.

Motivation & Objective

• To investigate whether isoclinic p-groups have stably isomorphic fields of invariants or isomorphic Bogomolov multipliers.
• To resolve a question posed in [4] about the invariance of the Bogomolov multiplier under isoclinism.
• To establish a structural connection between isoclinic groups and their unramified cohomology via the kernel of the commutator map.
• To provide a purely group-theoretic characterization of the Bogomolov multiplier that enables effective computation.

Proposed method

• Define the group $ G \curlywedge G $ as a quotient of the non-abelian exterior square, generated by symbols $ m \curlywedge n $ with specific relations involving conjugation and commutators.
• Introduce the group $ \tilde{B}_0(G) $ as the kernel of the commutator map $ \kappa: G \curlywedge G \to [G,G] $, which classifies the Bogomolov multiplier.
• Construct a $ \tilde{B}_0 $-pairing $ \phi: G_1 \times G_1 \to G_2 \curlywedge G_2 $ using the isoclinism data between $ G_1 $ and $ G_2 $, ensuring compatibility with the group operations.
• Show that this pairing induces a group isomorphism $ \gamma: G_1 \curlywedge G_1 \to G_2 \curlywedge G_2 $, preserving the commutator structure.
• Prove that the restriction of $ \gamma $ to $ \tilde{B}_0(G_1) $ yields an isomorphism $ \tilde{\gamma}: \tilde{B}_0(G_1) \to \tilde{B}_0(G_2) $, using the commutative diagram involving $ \beta $ and the commutator maps.
• Leverage the duality $ \operatorname{B}_0(G) \cong \operatorname{Hom}(\tilde{B}_0(G), \mathbb{Q}/\mathbb{Z}) $ to conclude that the Bogomolov multipliers are isomorphic.

Experimental results

Research questions

• RQ1 Do isoclinic finite groups have isomorphic Bogomolov multipliers?
• RQ2 Is the unramified Brauer group invariant under isoclinism, particularly in the context of Noether's problem?
• RQ3 Can the Bogomolov multiplier be described purely in terms of group-theoretic constructions like the non-abelian exterior square?
• RQ4 Does the isoclinism class of a p-group determine its rationality properties via the Bogomolov multiplier?
• RQ5 Is there a canonical isomorphism between the kernels of the commutator maps for isoclinic groups?

Key findings

• The kernel $ \tilde{B}_0(G) $ of the commutator map on $ G \curlywedge G $ is isomorphic for any two isoclinic groups $ G_1 $ and $ G_2 $, establishing a structural invariance.
• The isomorphism $ \tilde{\gamma}: \tilde{B}_0(G_1) \to \tilde{B}_0(G_2) $ is induced by the isoclinism data, specifically via the isomorphisms $ \alpha $ and $ \beta $, and preserves the group operation.
• The Bogomolov multiplier $ \operatorname{B}_0(G) $, defined as $ \operatorname{Hom}(\tilde{B}_0(G), \mathbb{Q}/\mathbb{Z}) $, is isomorphic for isoclinic finite groups.
• The construction is purely combinatorial and does not depend on the field or representation, making it suitable for algorithmic computation of the Bogomolov multiplier.
• The result confirms that the Bogomolov multiplier is an invariant of the isoclinism class of a finite group, answering a question posed in [4].
• The proof establishes a canonical isomorphism between the relevant subgroups of $ G_1 \curlywedge G_1 $ and $ G_2 \curlywedge G_2 $, preserving the commutator structure and the kernel of the commutator map.