Groups Acting Generically Multiply Transitively on Solvable Groups

2024/02/10 by Berkman, Ayşe, Borovik, Alexandre · 1 citation
#03C60 #20F11 #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO)

paper · doi:10.48550/arxiv.2402.06987

Abstract

In this work, we complete the classification of generically multiply transitive actions of groups on solvable groups in the finite Morley rank setting. We prove that if G is a connected group of finite Morley rank acting definably, faithfully and generically m-transitively on a connected solvable group V of finite Morley rank where rk(V)\leqslant m, then rk(V)=m, V is a vector space of dimension m over an algebraically closed field F, G≅ GL_m(F), and the action is equivalent to the natural action of GL_m(F) on F^m. This generalises our previous work arXiv:2107.09997. As an application of our result, we classify definably primitive groups of finite Morley rank and affine type acting on a set X with a generic transitivity degree of rk(X)+1.

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• The Şiş Kebap Theorem: A Generic Identification Theorem for Groups of Finite Morley Rank

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