Algebra and topology of factor-lattices

TL;DR

This paper explores the relationship between Abelian and non-Abelian groups of parity through the study of factor-lattices, revealing an isomorphism between their groups of automorphisms.

Contribution

It establishes a connection between Abelian and non-Abelian parity groups via the isomorphism of their automorphism groups in factor-lattices.

Findings

Isomorphism between automorphism groups of Abelian and non-Abelian parity groups

Characterization of parity groups as kernels of homomorphisms

Relation between regular lattice factorization and group structures

Abstract

In this paper we investigate the relation between Abelian and non-Abelian groups of parity. The Abelian groups of parity are formed as kernels of homomorphisms of parity in group $\mmathbb{Z}^{n}$ and the non-Abelian groups of parity are formed as kernels of homomorphisms of parity in group $S_{2}\wr S_{n}$. It is shown that the group of isomorphisms of the factor-lattice recieved by factorization of regular lattice with help of some Abelian group is equal to the corresponding non-Abelian group.