On Some arithmetic applications to the theory of symmetric groups

The work is devoted to some arithmetic applications to the theory of symmetric groups. Using the properties of congruences and classes of residues from number theory, the existence in the symmetric group $S_{n}$ of degree $n$ of cyclic, Abelian and non-Abelian subgroups respectively, of orders is establisned $k$, $\varphi(k)$, and $k \varphi(k)$, where $k \leq n$, $\varphi$ – Euler function, those representations jf grups $\left( \mathbb{Z} / k\mathbb{Z}, + \right)$, $\left( \mathbb{Z} / k\mathbb{Z} \right)^{*}$ and theorem product in the form of degree substitutions $k$. In this case isomorphic embeddings of these groups are constructed following the proof of Cayley's theorem, but along with this, a linear binomial is used $\mathbb{Z} / k\mathbb{Z}$ residue class rings, where $\gcd\left(a, k\right) = 1$.
In addition, the result concerning the isomorphic embedding of a group $\left( \mathbb{Z} / k\mathbb{Z} \right)^{*}$ in to a group $\left( \mathbb{Z} / k\mathbb{Z} \right)^{*}$ in to a group $S_{k}$ extends to an alternating group $A_{k}$ for odd $k$.
The second part of the work examines some applications of prime number theory to cyclic subgroups of the symmetric group $S_{n}$. In particular, applying the Euler-Maclaurin summation formula and bounds for the $k$ in prime, a lower bound for maximum number of prime divisors of cyclic orders in the summetric group $S_{n}$.