The Grunwald problem and homogeneous spaces with nonsolvable stabilisers

We give an affirmative answer to the Grunwald problem for new families of nonsolvable finite groups <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> , away from the set of primes dividing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>G</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> . Furthermore, we show that such <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> verify the condition (BM), that is, the Brauer–Manin obstruction to weak approximation is the only one for quotients of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mo form="prefix">SL</mml:mo> <mml:mi>n</mml:mi> </mml:msub> </mml:math> by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> . These new families include extensions of groups satisfying (BM) by kernels which are products of symmetric groups <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>𝔖</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:math> , with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:math> , and alternating groups <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>𝔄</mml:mi> <mml:mn>5</mml:mn> </mml:msub> </mml:math> . 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.