492 research outputs found
The minimal base size for a p-solvable linear group
Let be a finite vector space over a finite field of order and of
characteristic . Let be a -solvable completely reducible
linear group. Then there exists a base for on of size at most
unless in which case there exists a base of size at most . The
first statement extends a recent result of Halasi and Podoski and the second
statement generalizes a theorem of Seress. An extension of a theorem of P\'alfy
and Wolf is also given.Comment: 11 page
On a conjecture of Gluck
Let and respectively denote the Fitting subgroup and the
largest degree of an irreducible complex character of a finite group . A
well-known conjecture of D. Gluck claims that if is solvable then
. We confirm this conjecture in the case where
is coprime to 6. We also extend the problem to arbitrary finite groups and
prove several results showing that the largest irreducible character degree of
a finite group strongly controls the group structure.Comment: 16 page
A proof of Pyber's base size conjecture
Building on earlier papers of several authors, we establish that there exists a universal constant c>0 such that the minimal base size b(G) of a primitive permutation group G of degree n satisfies log|G|/logn≤b(G)1 we have the estimates |G|
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