Weakly Normal Groups

Abstract

Publisher Summary This chapter describes that a group G is weakly normal if and only if every definable X ⊂ G n is a Boolean combination of cosets of acl (ϕ)-definable subgroups ∀n if and only if every definable X ⊂ G n is a Boolean combination of cosets of definable subgroups (of G n ) ∀n. The chapter describes some equivalent conditions to that of weak normality and some basic results on stable groups. A locally connected definable subgroup of a weakly normal group is acl (ϕ)-definable. The chapter shows that if G is a weakly normal group, any p ∊ S(G) is determined by the definable cosets in p. The above result is enough to prove equivalence in the theorem, “an Abelian structure is an Abelian group A, together with distinguished subgroups of A n for various n's. Any Abelian structure is interpretable in a module.”