Abstract

Let $G$ be a $p$-group of maximal class and order $p^n$. We determine whether or not the Bogomolov multiplier ${\operatorname{B}}_0(G)$ 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 ${\operatorname{B}}_0(G)$ 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 non-trivial Bogomolov multipliers. We also provide more general results on Bogomolov multipliers of $p$-groups of arbitrary coclass $r$.

中文翻译：

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P类最大群的BOGOMOLOV乘数

摘要

假设$ G $是最大类的$ p $-组，并排序$ p ^ n $。我们确定Bogomolov乘数$ {\ operatorname {B}} _ 0（G）$在$ G $和$ P_1 = C_G（\ gamma _2（G）/ \ gamma _4的较低中央序列方面是否微不足道（G））$。如果另外，$ G $具有正交换性，并且$ P_1 $是阶变数，我们将展示对$ {\ operatorname {B}} _ 0（G）$的了解如何简化为$ P_1 $的简单换向器结构。该结果涵盖了足够大的阶的最大类的所有$ p $ -group，并且它使我们能够给出第一个自然的$ p $ -groups族，其中包含大量具有非平凡Bogomolov乘数的组。我们还提供了Bogomolov乘子的任意共类$ r $的$ p $-组的更一般的结果。