Subnormal and ascendant subgroups with rank restrictions (Q1114003)

Let \({\mathfrak X}\) denote the class of groups having finite abelian section rank, and let \(H_ 1,...,H_ n\) be subnormal \({\mathfrak X}\)-subgroups of the group G, with \(J=<H_ 1,...,H_ n>\). It is shown that J also belongs to \({\mathfrak X}\), and that J is ascendant in G. (J need not be subnormal.) An example is given which shows that the class of groups having finite (Prüfer) rank does not form on ``ascendant coalition class''.