CORRECTION OF A THEOREM ON THE SYMMETRIC GROUP GENERATED BY TRANSVECTIONS

DOI10.1093/QMATH/HAS014zbMATH Open1280.20052arXiv1108.2409OpenAlexW2963055093MaRDI QIDQ5300594

Publication date: 27 June 2013

Published in: The Quarterly Journal of Mathematics (Search for Journal in Brave)

Abstract: Let

V

denote a vector space over two-element field

m a t h b b F_{2}

with finite positive dimension and endowed with a symplectic form

B .

Let

m S L (V)

denote the special linear group of

V .

Let

S

denote a subset of

V .

Define

T v (S)

as the subgroup of

m S L (V)

generated by the transvections with direction

a l p h a

for all

a l p h a i n S .

Define

G (S)

as the graph whose vertex set is

S

and where

are connected whenever

A well-known theorem states that under the assumption that

S

spans

V,

the following (i), (ii) are equivalent: (i)

T v (S)

is isomorphic to a symmetric group. (ii)

G (S)

is a claw-free block graph. We give an example which shows that this theorem is not true. We give a modification of this theorem as follows. Assume that

S

is a linearly independent set of

V

and no element of

S

is in the radical of

V .

Then the above (i), (ii) are equivalent.

Full work available at URL: https://arxiv.org/abs/1108.2409

zbMATH Keywords

groups generated by transvections

Mathematics Subject Classification ID

Linear algebraic groups over finite fields (20G40) Generators, relations, and presentations of groups (20F05) Graphs and abstract algebra (groups, rings, fields, etc.) (05C25)