Finite \(n\)-coverings of groups (Q1340175)

An \(n\)-covering is meant as a generalization of a covering in the sense that every \(n\)-tuplet of elements is contained in one of the members of the \(n\)-covering. B. H. Neumann's Lemma on finite coverings has its analogue for \(n\)-coverings (Lemma 2.3). The group \(G\) possesses a finite non-trivial \(n\)-covering of subgroups (normal subgroups) if and only if \(G\) has a homomorphic image which is not generated by \(n\) elements and finite (elementary abelian), see section 2. Coverings by subgroups belonging to varieties are considered in their consequences to the group \(G\) itself in section 3, especially for properties near to nilpotency.