Schur's exponent conjecture -- counterexamples of exponent 5 and exponent 9

TL;DR

This paper provides counterexamples to Schur's exponent conjecture, showing it fails for groups of exponent 5 and 9 by exhibiting specific groups with larger Schur multiplier exponents.

Contribution

It presents the first known counterexamples to Schur's conjecture for exponents 5 and 9, expanding understanding of the conjecture's limitations.

Findings

Counterexample group of exponent 5 with Schur multiplier exponent 25

Counterexample group of exponent 9 with Schur multiplier exponent 27

Schur's conjecture fails for these specific exponents

Abstract

There is a long-standing conjecture attributed to I Schur that if $G$ is a finite group with Schur multiplier $M(G)$ then the exponent of $M(G)$ divides the exponent of $G$. It is easy to see that this conjecture holds for exponent 2 and exponent 3, but it has been known since 1974 that the conjecture fails for exponent 4. In this note I give an example of a group $G$ with exponent 5 with Schur multiplier $M(G)$ of exponent 25, and an example of a group $A$ of exponent 9 with Schur multiplier $M(A)$ of exponent 27.