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 of not the Bogomolov multiplier B 0 (G) is trivial in terms of the lower central series of G and P 1 = C G (γ 2 (G), γ 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 examples of p-groups with Bogomolov multiplier of arbitrarily high exponent. We also provide more general results on Bogomolov multipliers of p-groups of arbitrary coclass r. is in general negative, even when taking k = C. His approach was to use a certain Galois-cohomological invariant associated to the group G, namely, the unramified Brauer group. It was Bogomolov [3] who found a purely group-cohomological way of computing this invariant, now known as the Bogomolov multiplier and denoted by B 0 (G). This object represents an obstruction to Noether's problem, which has a negative answer for G provided that B 0 (G) is nontrivial.Recently, Moravec [22] introduced the curly exterior square G G, which is the group generated by the symbols x y for all pairs x, y ∈ G, subject to the relations xy z = (x y z y )(y z), x yz = (x z)(x z y z ), a b = 1, Date: October 31, 2018.

Some New Parallels Between Groups and Lie Algebras, or What Can Be Simpler than Multiplication Table?

“…Whereas the Bogomolov multiplier vanishes on finite simple groups [55] and simple Lie algebras (follows from the results of Peggy Batten's PhD thesis [6] where the vanishing of the Schur multiplier of simple Lie algebras is proven), on the other extreme edge (pgroups/nilpotent Lie algebras) there are nontrivial examples. For instance, in the paper [29] Gustavo Fernández-Alcober and Urban Jezernik showed that the Bogomolov multiplier of a p-group can be as large as we wish. In the papers [75,76] mentioned above, one can find examples of finite-dimensional nilpotent Lie algebras with nontrivial Bogomolov multiplier.…”

Irrationality of Generic Quotient Varieties via Bogomolov Multipliers

“…It follows from the above proofs that when the uniform subgroup of G is abelian, the group SK 1 (Z p G ) is of finite exponent bounded in terms of the rank of G, and so it is a finite p-group. This is the case, for example, when G is a pro-p group of finite coclass (see [12,Theorem 10.1] and [17,Theorem 4.4]).…”

On Schurs exponent conjecture and its relation to Noether's Rationality problem

“…Most p groups of maximal class have non-trivial Bogomolov multiplier (cf. [10]), but Conjecture 1 holds for p groups of maximal class. On the other hand, the Bogomolov multiplier is trivial for all cyclic groups, but Noether's Rationality problem does not hold for all cyclic groups.…”