Locally finite groups with all subgroups either subnormal or nilpotent-by-Chernikov. (Q424116)

Let \(F\) be an infinite locally finite field and \(K\) be an infinite subgroup of the multiplicative group \(F^*\) such that every infinite subgroup of \(K\) generates \(F\) as a ring. Then the extension of \(F^+\) by \(K\) is a group of the title. Whenever \(K\) is not a Chernikov group (and this can always be arranged), then \(F^+K\) is not nilpotent-by-Chernikov. These groups play a key role in the groups of the title with all \(p\)-sections being nilpotent-by-Chernikov for all primes \(p\) (Theorem 1.2). Groups of the title that are \(p\)-groups will be the subject of another article of the authors.