1,671 research outputs found
Invariable generation and the chebotarev invariant of a finite group
A subset S of a finite group G invariably generates G if G = <hsg(s) j s 2 Si
> for each choice of g(s) 2 G; s 2 S. We give a tight upper bound on the
minimal size of an invariable generating set for an arbitrary finite group G.
In response to a question in [KZ] we also bound the size of a randomly chosen
set of elements of G that is likely to generate G invariably. Along the way we
prove that every finite simple group is invariably generated by two elements.Comment: Improved versio
Braids: A Survey
This article is about Artin's braid group and its role in knot theory. We set
ourselves two goals: (i) to provide enough of the essential background so that
our review would be accessible to graduate students, and (ii) to focus on those
parts of the subject in which major progress was made, or interesting new
proofs of known results were discovered, during the past 20 years. A central
theme that we try to develop is to show ways in which structure first
discovered in the braid groups generalizes to structure in Garside groups,
Artin groups and surface mapping class groups. However, the literature is
extensive, and for reasons of space our coverage necessarily omits many very
interesting developments. Open problems are noted and so-labelled, as we
encounter them.Comment: Final version, revised to take account of the comments of readers. A
review article, to appear in the Handbook of Knot Theory, edited by W.
Menasco and M. Thistlethwaite. 91 pages, 24 figure
Conway groupoids, regular two-graphs and supersimple designs
A design is said to be supersimple
if distinct lines intersect in at most two points. From such a design, one can
construct a certain subset of Sym called a "Conway groupoid". The
construction generalizes Conway's construction of the groupoid . It
turns out that several infinite families of groupoids arise in this way, some
associated with 3-transposition groups, which have two additional properties.
Firstly the set of collinear point-triples forms a regular two-graph, and
secondly the symmetric difference of two intersecting lines is again a line. In
this paper, we show each of these properties corresponds to a group-theoretic
property on the groupoid and we classify the Conway groupoids and the
supersimple designs for which both of these two additional properties hold.Comment: 17 page
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