What question did this study set out to answer?

The central aim is to confirm the Grunwald problem for families of nonsolvable finite groups.

March 28, 2026Open Access

The Grunwald problem and homogeneous spaces with nonsolvable stabilisers

EBElyes BoughattasUniversity of Bath DNDanny NeftinTechnion – Israel Institute of Technology

Key Points

The central aim is to confirm the Grunwald problem for families of nonsolvable finite groups.
Analyzed conditions for nonsolvable groups away from prime divisors of |G|.
Demonstrated that these groups satisfy the Brauer–Manin condition (BM).
Provided explicit examples of small order groups related to (BM).
Investigated the interaction between (BM) and Schinzel’s hypothesis.
Affirmative answer to the Grunwald problem for new families of nonsolvable groups.
Verified that Brauer–Manin obstruction is unique for certain quotients of SLn by G.
Listed small order groups for which (BM) is unknown, with findings indicating that (BM) holds for many under specific conditions.

Abstract

We give an affirmative answer to the Grunwald problem for new families of nonsolvable finite groups G , away from the set of primes dividing | G | . Furthermore, we show that such G verify the condition (BM), that is, the Brauer–Manin obstruction to weak approximation is the only one for quotients of SL n by G . These new families include extensions of groups satisfying (BM) by kernels which are products of symmetric groups 𝔖 m , with m ≠ 2 , 6 , and alternating groups 𝔄 5 . We also investigate (BM) for small groups by giving an explicit list of small order groups for which (BM) is unknown and we show that for many of them (BM) holds under Schinzel’s hypothesis.

Demander à l'IA

Bookmark

View Full Paper