101 research outputs found
Bases of quasisimple linear groups
Let be a vector space of dimension over , a finite field of
elements, and let be a linear group. A base of
is a set of vectors whose pointwise stabiliser in is trivial. We prove that
if is a quasisimple group (i.e. is perfect and is simple)
acting irreducibly on , then excluding two natural families, has a base
of size at most 6. The two families consist of alternating groups
acting on the natural module of dimension or , and classical
groups with natural module of dimension over subfields of
Finite subgroups of simple algebraic groups with irreducible centralizers
We determine all finite subgroups of simple algebraic groups that have
irreducible centralizers - that is, centralizers whose connected component does
not lie in a parabolic subgroup.Comment: 24 page
Recognition of finite exceptional groups of Lie type
Let be a prime power and let be an absolutely irreducible subgroup of
, where is a finite field of the same characteristic as \F_q,
the field of elements. Assume that , a quasisimple group of
exceptional Lie type over \F_q which is neither a Suzuki nor a Ree group. We
present a Las Vegas algorithm that constructs an isomorphism from to the
standard copy of . If with even, then the
algorithm runs in polynomial time, subject to the existence of a discrete log
oracle
The length and depth of algebraic groups
Let be a connected algebraic group. An unrefinable chain of is a
chain of subgroups , where each is a
maximal connected subgroup of . We introduce the notion of the length
(respectively, depth) of , defined as the maximal (respectively, minimal)
length of such a chain. Working over an algebraically closed field, we
calculate the length of a connected group in terms of the dimension of its
unipotent radical and the dimension of a Borel subgroup of the
reductive quotient . In particular, a simple algebraic group of rank
has length , which gives a natural extension of a theorem of
Solomon and Turull on finite quasisimple groups of Lie type. We then deduce
that the length of any connected algebraic group exceeds .
We also study the depth of simple algebraic groups. In characteristic zero,
we show that the depth of such a group is at most (this bound is sharp). In
the positive characteristic setting, we calculate the exact depth of each
exceptional algebraic group and we prove that the depth of a classical group
(over a fixed algebraically closed field of positive characteristic) tends to
infinity with the rank of the group.
Finally we study the chain difference of an algebraic group, which is the
difference between its length and its depth. In particular we prove that, for
any connected algebraic group , the dimension of is bounded above
in terms of the chain difference of .Comment: 18 pages; to appear in Math.
Base sizes for simple groups and a conjecture of Cameron
Let G be a permutation group on a finite set ?. A base for G is a subset B C_ ? whose pointwise stabilizer in G is trivial; we write b(G) for the smallest size of a base for G. In this paper we prove that b(G) ? if G is an almost simple group of exceptional Lie type and is a primitive faithful G-set. An important consequence
of this result, when combined with other recent work, is that b(G) ? 7 for any almost simple group G in a non-standard action, proving a conjecture of Cameron. The proof is probabilistic and uses bounds on fixed point ratios
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