67 research outputs found
Primitive permutation groups and derangements of prime power order
Let be a transitive permutation group on a finite set of size at least
. By a well known theorem of Fein, Kantor and Schacher, contains a
derangement of prime power order. In this paper, we study the finite primitive
permutation groups with the extremal property that the order of every
derangement is an -power, for some fixed prime . First we show that these
groups are either almost simple or affine, and we determine all the almost
simple groups with this property. We also prove that an affine group has
this property if and only if every two-point stabilizer is an -group. Here
the structure of has been extensively studied in work of Guralnick and
Wiegand on the multiplicative structure of Galois field extensions, and in
later work of Fleischmann, Lempken and Tiep on -semiregular pairs.Comment: 30 pages; to appear in Manuscripta Mat
Covers and Normal Covers of Finite Groups
For a finite non cyclic group , let be the smallest integer
such that contains proper subgroups with the
property that every element of is contained in for some and We prove that if is a noncyclic permutation
group of degree then We then investigate the
structure of the groups with (where is
the size of a minimal cover of ) and of those with $\gamma(G)=2.
Product decompositions of quasirandom groups and a Jordan type theorem
We first note that a result of Gowers on product-free sets in groups has an
unexpected consequence: If k is the minimal degree of a representation of the
finite group G, then for every subset B of G with we have
B^3 = G.
We use this to obtain improved versions of recent deep theorems of Helfgott
and of Shalev concerning product decompositions of finite simple groups, with
much simpler proofs.
On the other hand, we prove a version of Jordan's theorem which implies that
if k>1, then G has a proper subgroup of index at most ck^2 for some absolute
constant c, hence a product-free subset of size at least . This
answers a question of Gowers.Comment: 18 pages. In this third version we added an Appendix with a short
proof of Proposition
- …