Normal subgroups of iterated wreath products of symmetric groups and alternating with symmetric groups

Normal subgroups and there properties for finite and infinite iterated wreath products $S_{n_1}\wr \ldots \wr S_{n_m}$, $n, m \in \mathbb{N}$ are founded. The special classes of normal subgroups and there orders are investigated. Special classes of normal subgroups are investigated and their generators are found and presented in the form of Kaloujnine tables. Inverse limit of wreath product of permutation groups is found. Conditions when set-wise stabilizer is normal subgroup are found. In the proposed work a topology of the wreath product as profinite group is studied, is homeomorphic to Cantor set is established.