Finite 2-groups with index of every cyclic subgroup in its normal closure no greater than 4. (Q661382)

A \(2\)-group \(G\) is said to be \(\text{BI}(2^2)\)-group if \(|C^G:C|\leq 2^2\) for all cyclic \(C<G\) (here \(C^G\) is the normal closure of \(C\) in \(G\)). A number of non-trivial properties of \(\text{BI}(2^2)\)-groups are proved. However, this class of \(2\)-groups is not classified. A similar problem for \(p>2\) was considered in great detail by Z. Janko in \S144 of the book [\textit{Y. Berkovich} and \textit{Z. Janko}, Groups of prime power order. Vol. 3. Berlin: Walter de Gruyter (2011; Zbl 1229.20001)].