Bogomolov multipliers of $p$-groups of maximal class

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 $r$.