arXiv subjects
Publications and source records attributed to Ayşe Berkman.
This paper contains a new version of a final identification theorem for the `generic' groups of finite Morley rank, developing further the results of [4] and [5]. It is prepared for use in the Hrushovski Programme of classification of infinite simple groups of finite Morley rank.
We give a review of one of the lines in development of the theory of groups of finite Morley rank. These groups naturally appear in model theory as model-theoretic analogues of Galois groups, therefore their actions and their role as permutation groups is of primary interest. We restrict our story to the study of connected groups of finite Morley rank $G$ acting in a definably primitive way on a set $X$ and containing a definable abelian normal subgroup $V$ which acts on $X$ regularly -- the so-called \emph{primitive groups of affine type}. For reasons explained in the paper, this case plays a central role in the theory.
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 $\operatorname{rk}(V)\leqslant m$, then $\operatorname{rk}(V)=m$, $V$ is a vector space of dimension $m$ over an algebraically closed field $F$, $G\cong \operatorname{GL}_m(F)$, and the action is equivalent to the natural action of $\operatorname{GL}_m(F)$ 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 $\operatorname{rk}(X)+1$.
We investigate the configuration where a group of finite Morley rank acts definably and generically $m$-transitively on an elementary abelian $p$-group of Morley rank $n$, where $p$ is an odd prime, and $m\geqslant n$. We conclude that $m=n$, and the action is equivalent to the natural action of $\operatorname{GL}_n(F)$ on $F^n$ for some algebraically closed field $F$. This strengthens our earlier result in arXiv:1802.05222, and partially answers two problems posed in [9].
We prove that if $G$ is a group of finite Morley rank which acts definably and generically sharply $n$-transitively on a connected abelian group $V$ of Morley rank $n$ with no involutions, then there is an algebraically closed field $F$ of characteristic $\ne 2$ such that $V$ has a structure of a vector space of dimension $n$ over $F$ and $G$ acts on $V$ as the group $\operatorname{GL}_n(F)$ in its natural action on $F^n$. This is the final pre-publication version of the paper: A. Berkman and A. Borovik, Groups of finite Morley rank with a generically sharply multiply transitive action, J. Algebra (2018), https://doi.org/10.1016/j.jalgebra.2018.07.033. Accepted for publication 28 July 2018. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published