Examples for Bogomolov multiplier of finite group

Question

Take $G$ to be a finite group. The projective representations of $G$ are classified by the group cohomology $H^2(G,U(1))$. Here $G$ has a trivial action on $U(1)$. We focus on the restriction map

$H^2(G,U(1)) \xrightarrow[]{\rm restriction} \oplus_{B\subseteq G}~ H^2(B,U(1))$,

where $\oplus_{B\subseteq G}$ means the direct sum over all bicyclic subgroups $B = Z_m × Z_n$ of $G$. The kernel of this restriction map is defined as the Bogomolov multiplier $B_0(G)$ of $G$, namely

$B_0(G):={\rm Ker}(~H^2(G,U(1)) \xrightarrow[]{\rm restriction} \oplus_{B\subseteq G}~ H^2(B,U(1))~) $.

My question is what is the simplest finite group $G$ with a non-trivial $B_0(G)$? And how do I express the corresponding non-trivial element of $B_0(G)$ (as a function $G \times G \rightarrow U(1)$)?

Answer 1

I realize that I'm chiming into a rather old question here, but it turns out that I wanted to know the answer myself awhile ago, and it seems that there's an interesting story lurking here still.

To cut to the chase: the smallest examples of groups with non-trivial Bogomolov multipliers have order $64=2^6$. In fact, there are 9 such groups, and they can be called from the GAP SmallGroup library using $\text{SmallGroup}(64,i)$ where $$i \in \{149, 150, 151, 170, 171, 172, 177, 178, 182\}.$$ If my memory is correct, then they are all isoclinic. It's maybe worth noting that the smallest examples must be weird $p$-groups, because the Bogomolov multiplier vanishes on abelian groups (easy), as well as simple groups (hard-it uses the classification! This result is due to Kunyavskii.).

For some context: I found these examples in the course of the work in equivariant bordism that I did here, although that was before I had heard of the phrase "Bogomolov multiplier" and so I had to find them by brute force using my own functions in GAP. It was only after reading Primoz Moravec's nice paper that I realized that my bordism theoretic formulation and the algebraic ones (both homological and cohomological) are all isomorphic, and so these examples were already known. (Moreover, in retrospect, I realized that if you use the HAP package in GAP, then you can just call the function BogomolovMultiplier! However, because I needed to quickly rule out lots of groups with trivial Bogomolov multiplier, I baked in some randomization methods that seemed to speed things up compared to HAP.)

I say all of this because it's still rather mysterious to me why there would be much connection between equivariant bordism with branch loci (which is what I was studying), the Noether problem over $\mathbb{C}$ (which is what Saltman and Bogomolov were studying, and a motivation for Moravec), and braided auto-equivalences of $DG$-mod (which is what Davydov was studying, and presumably what brought you here :-) )