791 research outputs found
Large dimensional classical groups and linear spaces
Suppose that a group has socle a simple large-rank classical group.
Suppose furthermore that acts transitively on the set of lines of a linear
space . We prove that, provided has dimension at least 25,
then acts transitively on the set of flags of and hence the
action is known. For particular families of classical groups our results hold
for dimension smaller than 25.
The group theoretic methods used to prove the result (described in Section 3)
are robust and general and are likely to have wider application in the study of
almost simple groups acting on finite linear spaces.Comment: 32 pages. Version 2 has a new format that includes less repetition.
It also proves a slightly stronger result; with the addition of our
"Concluding Remarks" section the result holds for dimension at least 2
A Census Of Highly Symmetric Combinatorial Designs
As a consequence of the classification of the finite simple groups, it has
been possible in recent years to characterize Steiner t-designs, that is
t-(v,k,1) designs, mainly for t = 2, admitting groups of automorphisms with
sufficiently strong symmetry properties. However, despite the finite simple
group classification, for Steiner t-designs with t > 2 most of these
characterizations have remained longstanding challenging problems. Especially,
the determination of all flag-transitive Steiner t-designs with 2 < t < 7 is of
particular interest and has been open for about 40 years (cf. [11, p. 147] and
[12, p. 273], but presumably dating back to 1965). The present paper continues
the author's work [20, 21, 22] of classifying all flag-transitive Steiner
3-designs and 4-designs. We give a complete classification of all
flag-transitive Steiner 5-designs and prove furthermore that there are no
non-trivial flag-transitive Steiner 6-designs. Both results rely on the
classification of the finite 3-homogeneous permutation groups. Moreover, we
survey some of the most general results on highly symmetric Steiner t-designs.Comment: 26 pages; to appear in: "Journal of Algebraic Combinatorics
Scott's formula and Hurwitz groups
This paper continues previous work, based on systematic use of a formula of
L. Scott, to detect Hurwitz groups. It closes the problem of determining the
finite simple groups contained in for which are Hurwitz,
where is an algebraically closed field. For the groups , ,
and the Janko groups and it provides explicit -generators
Linear spaces with significant characteristic prime
Let be a group with socle a simple group of Lie type defined over the
finite field with elements where is a power of the prime . Suppose
that acts transitively upon the lines of a linear space . We
show that if is {\it significant} then acts flag-transitively on
and all examples are known.Comment: 11 page
Finite projective planes admitting a projective linear group PSL (2,q)
AbstractLet S be a projective plane, and let G⩽Aut(S) and PSL(2,q)⩽G⩽PΓL(2,q) with q>3. If G acts point-transitively on S, then q=7 and S is of order 2
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