# Parametrized symmetric groups and the second homology of a group

## TL;DR

This paper introduces parametrized symmetric groups linked to a given group and explores their structure as extensions involving the second homology group, providing new insights into group extensions and homology.

## Contribution

It defines parametrized symmetric groups and demonstrates their role as extensions of specific subgroups by the second homology group, advancing understanding of group homology and extensions.

## Key findings

- Parametrized symmetric groups are extensions of certain wreath product subgroups by H_2(G, Z).
- The construction offers a new perspective on the relationship between symmetric groups and group homology.
- The paper discusses the motivation and potential applications of these groups in algebraic topology.

## Abstract

We introduce the notion of a symmetric group parametrized by elements of a group. We show that this group is an extension of a certain subgroup of the wreath product $G \wr S_n$ by $\mathrm{H}_2(G, \mathbb{Z})$. We also discuss the motivation behind this construction.