157 research outputs found
Finitely generated groups with polynomial index growth
We prove that a finitely generated soluble residually finite group has
polynomial index growth if and only if it is a minimax group. We also show that
if a finitely generated group with PIG is residually finite-soluble then it is
a linear group.
These results apply in particular to boundedly generated groups; they imply
that every infinite BG residually finite group has an infinite linear quotient.Comment: To appear in Crelle's Journa
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
Normalizers of Primitive Permutation Groups
Let be a transitive normal subgroup of a permutation group of finite
degree . The factor group can be considered as a certain Galois group
and one would like to bound its size. One of the results of the paper is that
if is primitive unless , , , , or
. This bound is sharp when is prime. In fact, when is
primitive, unless is a member of a given infinite
sequence of primitive groups and is different from the previously listed
integers. Many other results of this flavor are established not only for
permutation groups but also for linear groups and Galois groups.Comment: 44 pages, grant numbers updated, referee's comments include
Some simplifications in the proof of the Sims conjecture
We prove an elementary lemma concerning primitive amalgams and use it to
greatly simplify the proof of the Sims conjecture in the case of almost simple
groups
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