Hall's criterion for nilpotence in semi-abelian categories

Abstract

A well-known theorem of P. Hall, usually called Hall's criterion for nilpotence, states: a group G is nilpotent whenever it has a normal subgroup N such that G/[N,N] and N are nilpotent. We widely generalize this result, replacing groups with objects in an abstract semi-abelian category satisfying suitable conditions. In particular, these conditions are satisfied in any algebraically coherent semi-abelian category, and hence in many categories of classical algebraic structures (including a few where Hall's theorem is already known). Note that the categories of crossed modules, crossed Lie algebras, n-cat groups, and cocommutative Hopf algebras over fields are also algebraically coherent. We give, however, a counter-example showing that Hall's theorem does not hold in the (semi-abelian) variety of non-associative rings.