Schur's exponent conjecture II

TL;DR

This paper advances the understanding of Schur's exponent conjecture by providing bounds and computational results for the exponent of the Schur multiplier in finite groups of specific exponents and nilpotency classes, extending Moravec's 2007 work.

Contribution

It refines bounds on the exponent of Schur multipliers for finite groups, computes specific cases for groups of exponent 8 and 9, and explores the order of commutators in Schur covers.

Findings

Bound e ≥ n for groups of exponent n=8,9.

e_{p^{k},p^{2}-p-1} divides p for all prime powers p^{k}.

e_{2^{k},c} and e_{3^{k},c} have specific values depending on c.

Abstract

Primoz Moravec published a very important paper in 2007 where he proved that if $G$ is a finite group of exponent $n$ then the exponent of the Schur multiplier of $G$ can be bounded by a function $f(n)$ depending only on $n$. Moravec does not give a value for $f(n)$, but actually his proof shows that we can take $f(n)=ne$ where $e$ is the order of $b^{-n}a^{-n}(ab)^{n}$ in the Schur multiplier of $R(2,n)$. (Here$R(2,n)$ is the largest finite two generator group of exponent $n$, and we take $a,b$ to be the generators of $R(2,n)$.) It is an easy hand calculation to show that $e=n$ for $n=2,3$, and it is a straightforward computation with the $p$-quotient algorithm to show that $e=n$ for $n=4,5,7$. The groups $R(2,8)$ and $R(2,9)$ are way out of range of the $p$-quotient algorithm, even with a modern supercomputer. But we are able toshow that $e\geq n$ for $n=8,9$. Moravec's proof also shows that if $G$ is a finite group of exponent $n$ with nilpotency class $c$, then the exponent of the Schur multiplier of $G$ is bounded by $ne$ where $e$ is the order of $b^{-n}a^{-n}(ab)^{n}$ in the Schur multiplier of the class $c$ quotient $R(2,n;c)$ of $R(2,n)$. If $q$ is a prime power we let $e_{q,c}$ be the order of $b^{-q}a^{-q}(ab)^{q}$ in the Schur multiplier of $R(2,q;c)$. We are able to show that $e_{p^{k},p^{2}-p-1}$ divides $p$ for all prime powers $p^{k}$. If $k>2$ then $e_{2^{k},c}$ equals 2 for $c<4$, equals 4 for $4\leq c\leq11$, and equals $8$ for $c=12$. If $k>1$ then $e_{3^{k},c}$ equals 1 for $c<3$, equals 3 for $3\leq c<12$, and equals 9 for $c=12$. We also investigate the order of $[b,a]$ in a Schur cover for $R(2,q;c)$.