arXiv subjects

R. M. Guralnick

Publications and source records attributed to R. M. Guralnick.

4 recordsLinked to original sources

arXiv2025

Generic stabilizers in actions of simple algebraic groups

In this paper we treat faithful actions of simple algebraic groups on irreducible modules and on the associated Grassmannian varieties. By explicit calculation, we show that in each case, with essentially one exception (only in characteristic 2), there is a dense open subset any point of which has stabilizer conjugate to a fixed subgroup, called the generic stabilizer. We provide tables listing generic stabilizers in the cases where they are non-trivial; in addition we decide whether or not there is a dense orbit, or a regular orbit for the action on the module.

math.GR↗

arXiv2021

Generic stabilizers in actions of simple algebraic groups II: higher Grassmannian varieties

This is the second of two papers treating faithful actions of simple algebraic groups on irreducible modules and on the associated Grassmannian varieties; in the first paper we considered the module itself and its projective space, while here we handle the varieties comprising the subspaces of the module of fixed dimension greater than one. By explicit calculation, we show that in each case, with essentially one exception, there is a dense open subset any point of which has stabilizer conjugate to a fixed subgroup, called the generic stabilizer. We provide tables listing generic stabilizers in the cases where they are non-trivial; in addition we decide whether or not there is a dense orbit.

math.GR↗

arXiv2009

Remarks on Proficient groups

If a finite group G has a presentation with d generators and r relations, it is well-known that r - d is at least the rank of the Schur multiplier of G; a presentation is called efficient if equality holds. There is an analogous definition for proficient profinite presentations. We show that many perfect groups have proficient presentations. Moreover, we prove that infinitely many alternating groups, symmetric groups and their double covers have proficient presentations

math.GR↗

arXiv2008

Presentations of finite simple groups: a computational approach

All nonabelian finite simple groups of rank $n$ over a field of size $q$, with the possible exception of the Ree groups $^2G_2(3^{2e+1})$, have presentations with at most $80 $ relations and bit-length $O(\log n +\log q)$. Moreover, $A_n$ and $S_n$ have presentations with 3 generators$,$ 7 relations and bit-length $O(\log n)$, while $\SL(n,q)$ has a presentation with 7 generators, $2 5$ relations and bit-length $O(\log n +\log q)$.

math.GR↗