BOGOMOLOV MULTIPLIERS OF P-GROUPS OF MAXIMAL CLASS

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 $r$.

On the exponent of Bogomolov multipliers

We prove that if G is a finite group, then the exponent of its Bogomolov multiplier divides the exponent of G in the following four cases: (i) G is metabelian, (ii) {\exp G=4} , (iii) G is nilpotent of class {\leq 5} , or (iv) G is a 4-Engel group.

Commutativity preserving extensions of groups

In parallel to the classical theory of central extensions of groups, we develop a version for extensions that preserve commutativity. It is shown that the Bogomolov multiplier is a universal object parametrizing such extensions of a given group. Maximal and minimal extensions are inspected, and a connection with commuting probability is explored. Such considerations produce bounds for the exponent and rank of the Bogomolov multiplier.

Units of group rings, the Bogomolov multiplier and the fake degree conjecture

Let π be a finite p-group and ${\mathbb{F}_{q}}$ a finite field with q = pn elements. Denote by $\I_{\mathbb{F}_{q}}$ the augmentation ideal of the group ring ${\mathbb{F}_{q}}$[π]. We have found a surprising relation between the abelianization of 1 + $\I_{\mathbb{F}_{q}}$, the Bogomolov multiplier B0(π) of π and the number of conjugacy classes k(π) of π: $$ \left | (1+\I_{\Fq})_{\ab} \right |=q^{\kk(\pi)-1}|\!\B_0(\pi)|. In particular, if π is a finite p-group with a non-trivial Bogomolov multiplier, then 1 + $\I_{\mathbb{F}_{q}}$ is a counterexample to the fake degree conjecture proposed by M. Isaacs.