# Identity theorem for pro-$p$-groups

## TL;DR

This paper proves the Identity Theorem for pro-$p$-groups with one defining relation, addressing a question by Serre and introducing 'conjurings' to demonstrate the result's finality.

## Contribution

It establishes the Identity Theorem for pro-$p$-groups with a single relation, providing new insights into their structure and relation modules.

## Key findings

- Proves the Identity Theorem for pro-$p$-groups with one relation
- Addresses a question posed by Serre on relation modules
- Introduces 'conjurings' to show the theorem's finality

## Abstract

We prove the Identity Theorem for pro-$p$-groups with a single defining relation giving a positive feedback to a question of Serre on the structure of relation modules. A construction of "conjurings" indicates finality of our result in a certain sense.