On Relative Central Extensions and Covering Pairs

TL;DR

This paper introduces a construction for relative central extensions of group pairs, explores covering pairs of perfect pairs, and extends properties of perfect groups to perfect pairs, providing new insights into their structure.

Contribution

It constructs a relative central extension for group pairs, characterizes covering pairs of perfect pairs, and extends properties of perfect groups to perfect pairs.

Findings

Every perfect pair admits at least one covering pair.

Special types of covering pairs are homomorphic images of the constructed extension.

Properties of perfect groups are extended to perfect pairs under certain conditions.

Abstract

Let $(G,N)$ be a pair of groups. In this article, first we construct a relative central extension for the pair $(G,N)$ such that special types of covering pair of $(G,N)$ are homomorphic image of it. Second, we show that every perfect pair admits at least one covering pair. Finally, among extending some properties of perfect groups to perfect pairs, we characterize covering pairs of a perfect pair $(G,N)$ under some extra assumptions.