Actions of 𝐴𝑙𝑡(𝑛) on groups of finite Morley rank without involutions

Abstract

We investigate faithful representations of Alt ⁡ ( n ) \operatorname {Alt}(n) as automorphisms of a connected group G G of finite Morley rank. We target a lower bound of n n on the rank of such a nonsolvable G G , and our main result achieves this in the case when G G is without involutions. In the course of our analysis, we also prove a corresponding bound for solvable G G by leveraging recent results on the abelian case. We conclude with an application towards establishing natural limits to the degree of generic transitivity for permutation groups of finite Morley rank.