Schur’s exponent conjecture

Abstract: If $G$ is a finite group and we write $G = F/R$ where $F$ is a free group, then the Schur multiplier $M(G)$ is $(R \cap F')/[F, R]$.

There is a long-standing conjecture attributed to I. Schur that the exponent of $M(G)$ divides the exponent of $G$. It is easy to show that this is true for groups $G$ of exponent $2$ or exponent $3$, but it has been known since 1974 that the conjecture fails for exponent $4$. However the truth or otherwise of this conjecture has remained open up till now for groups of odd exponent.

In my talk I describe counterexamples to the conjecture of exponent $5$ and exponent $9$.

I also give some suggestions for further counterexamples, and explore the possibilities for alternative conjectures.

group theory

Audience: researchers in the topic

GOThIC - Ischia Online Group Theory Conference

Series comments: Please send a message to andrea.caranti@unitn.it to receive a link to the Zoom room where the conference (a series of talks, actually) takes place.