Explicit inverse Shapiro isomorphism and its application

TL;DR

This paper derives an explicit inverse of Shapiro's lemma using inhomogeneous cocycles and applies it to construct specific nonsplit group coverings with unique involution conjugacy classes.

Contribution

It provides a concrete formula for the inverse Shapiro isomorphism and demonstrates its application in group theory constructions.

Findings

Explicit inverse form of Shapiro's lemma derived.

Constructed special nonsplit group coverings with unique involution class.

Enhanced understanding of cocycle-based isomorphisms in group theory.

Abstract

We find an explicit form of the inverse isomorphism from Shapiro's lemma in terms of inhomogeneous cocycles and apply it to construct special nonsplit coverings of groups with a unique conjugacy class of involutions.