299 research outputs found
A classification of primitive permutation groups with finite stabilizers
We classify all infinite primitive permutation groups possessing a finite
point stabilizer, thus extending the seminal Aschbacher-O'Nan-Scott Theorem to
all primitive permutation groups with finite point stabilizers.Comment: Accepted in J. Algebra. Various changes, some due to the author, some
due to suggestions from readers and others due to the comments of anonymous
referee
Groups of type via graphical small cancellation
We construct an uncountable family of groups of type . In contrast to
every previous construction of non-finitely presented groups of type we do
not use Morse theory on cubical complexes; instead we use Gromov's graphical
small cancellation theory.Comment: 3 figures. Second version: two paragraphs added emphasizing the
difference between our construction and Morse theoretic one
Pariah moonshine
Finite simple groups are the building blocks of finite symmetry. The effort
to classify them precipitated the discovery of new examples, including the
monster, and six pariah groups which do not belong to any of the natural
families, and are not involved in the monster. It also precipitated monstrous
moonshine, which is an appearance of monster symmetry in number theory that
catalysed developments in mathematics and physics. Forty years ago the pioneers
of moonshine asked if there is anything similar for pariahs. Here we report on
a solution to this problem that reveals the O'Nan pariah group as a source of
hidden symmetry in quadratic forms and elliptic curves. Using this we prove
congruences for class numbers, and Selmer groups and Tate--Shafarevich groups
of elliptic curves. This demonstrates that pariah groups play a role in some of
the deepest problems in mathematics, and represents an appearance of pariah
groups in nature.Comment: 20 page
The (2,3)-generation of the classical simple groups of dimension 6 and 7
In this paper we prove that the finite simple groups ,
and are (2,3)-generated for all q. In particular,
this result completes the classification of the (2,3)-generated finite
classical simple groups up to dimension 7
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