Bogomolov multipliers of $p$-groups of maximal class

TL;DR

This paper investigates the Bogomolov multipliers of $p$-groups of maximal class, providing criteria for triviality, reducing complexity under certain conditions, and exploring their behavior across different coclasses.

Contribution

It offers new criteria for the triviality of Bogomolov multipliers in $p$-groups of maximal class and introduces the first natural family of such groups with nontrivial multipliers.

Findings

Criteria for trivial Bogomolov multipliers based on the lower central series.

Reduction of the problem to simpler commutator structures when $P_1$ is metabelian.

Existence of a natural family of $p$-groups with nontrivial Bogomolov multipliers.

Abstract

Let $G$ be a $p$-group of maximal class and order $p^n$. We determine whether or not the Bogomolov multiplier $B_0(G)$ is trivial in terms of the lower central series of $G$ and $P_1 = C_G(\gamma_2(G) / \gamma_4(G))$. If in addition $G$ has positive degree of commutativity and $P_1$ is metabelian, we show how understanding $B_0(G)$ reduces to the simpler commutator structure of $P_1$. This result covers all $p$-groups of maximal class of large enough order and, furthermore, it allows us to give the first natural family of $p$-groups containing an abundance of groups with nontrivial Bogomolov multipliers. We also provide more general results on Bogomolov multipliers of $p$-groups of arbitrary coclass $r$.