Properties of the group generated by translation groups of the vector space and the residue ring

In this paper, we consider the additive group $\mathbb Z_{2^n}^+$ of the residue ring $\mathbb Z_{2^n}$, the additive group $V_n^+$ of the vector space $V_n$ over the field $\mathrm{GF}(2)$, and subgroups of the group $G_n$ generated by $\mathbb Z_{2^n}^+$, $V_n^+$. These groups are subgroups of the Sylow $2$-subgroup of the symmetrical group $S(\mathbb Z_{2^n})$ and have common systems of imprimitivity. In cryptography, $\mathbb Z_{2^n}^+$, $V_n^+$ are connected with groups generated by all key additions. We describe a permutation structure of subgroups of $G_n$. We prove that the group of lower triangular $(n\times n)$-matrices over $\mathrm{GF}(2)$ and the full affine group over $\mathbb Z_{2^n}$ are subgroups of ${G_n}$. We also describe properties of imprimitive subgroups of $G_n$.