In the paper of Duke and Hopkins (2005), following the approach of E.I. Zolotarev, an analogue of the quadratic reciprocity law for groups was obtained using the Kronecker symbol. We present a short proof of this statement using the Jacobi symbol. The work is mainly of a methodological nature. In this regard, we also provide a proof of the result established in the paper by Frobenius (1914), related to the combinatorial interpretation of the Jacobi symbol.

We construct an algorithm for transforming words using a set of finite quasigroups in an amount equal to the number of characters of the alphabet. Some properties of ternary (L,M)-quasigroups are given, which play an important role in the analysis and design of cryptographic schemes based on these algebras, such as polynomial completeness, absence of nontrivial congruences.

A rigid isotopy of real algebraic curves of a certain class is a path in the space of curves of this class. Our study completes the rigid isotopic classification of nonsingular real algebraic curves of bidegree (4,3) on a hyperboloid, started by the author in his earlier works. Missing proofs of the uniqueness of the connected components for 16 classes of real algebraic curves of bidegree (4,3) having a single node or a cusp are given. The main technical tools are graphs of real trigonal curves on Hirzebruch surfaces.

We prove that if the compressed zero-divisor graph of a finite associative ring contains only one strong vertex then this vertex has a loop. Moreover, more properties of the compressed zero-divisor graph of a finite associative ring are proved.

The Levi class L(M) generated by the class of groups M is the class of all groups in which the normal closure of each cyclic subgroup belongs to M.

Let p be a prime number, p ̸= 2, s be a natural number, s ≥ 2, and s > 2 for p = 3; H_ps be a free group of rank 2 in the variety of nilpotent groups of class ≤ 2 of exponent p^{s with commutator subgroup of exponent p; Z is an infinite cyclic group; q{H}_p^{s , Z} is a quasivariety generated by the set of groups {H}_p^{s , Z}. We find a basis of quasi-identities of the Levi class L(q{H}_p^{s , Z}) and establish that there exists a continuous set of quasivarieties K such that L(K) = L(q{H}_p^{s , Z}).}