Imprimitivity systems and lattices of normal subgroups in $D$-hyperoctahedral groups

We study $D$-hyperoctahedral groups, diagonal inductive limits of hyperoctahedral groups. Also, we describe the $Z_2$-modules of periodic sequences over diagonal limits of symmetric groups under their action on the elements of a $Z_2$-module by permutation of coordinates. The imprimitivity systems for $D$-hyperoctahedral groups are characterized, and a full description of the lattices of their normal subgroups is given.