Groups rich with almost-normal subgroups (Q1123976)

H is almost normal in G if \(| G:N_ G(H)| =| Cl_ G(H)|\) is finite. The authors consider the minimal condition on subgroups which are not almost normal (Min-\(\overline{an})\). A group satisfying this condition can be considered to have many almost normal subgroups. Theorem 1: A nonperiodic group G satisfies Min-\(\overline{an}\) if and only if it is centre-by-finite or it has a finite normal subgroup F such that \(G/F\cong K\times (A\leftthreetimes <b>)\) where K is divisible Chernikov, A is free abelian of rank p-1 and b is an element of order p acting rationally irreducibly on A. Theorem 2: A locally finite group G satisfies Min-\(\overline{an}\) if and only if it is Chernikov or centre-by-finite.