1,335 research outputs found
On the regularity of a graph related to conjugacy classes of groups
AbstractGiven a finite group G, denote by Γ(G) the simple undirected graph whose vertices are the (distinct) non-central conjugacy class sizes of G, and for which two vertices of Γ(G) are adjacent if and only if they are not coprime numbers. In this note we prove that Γ(G) is a 2-regular graph if and only if it is a complete graph with three vertices, and Γ(G) is a 3-regular graph if and only if it is a complete graph with four vertices
Group actions on 1-manifolds: a list of very concrete open questions
This text focuses on actions on 1-manifolds. We present a (non exhaustive)
list of very concrete open questions in the field, each of which is discussed
in some detail and complemented with a large list of references, so that a
clear panorama on the subject arises from the lecture.Comment: 21 pages, 2 figure
Generic Newton points and the Newton poset in Iwahori double cosets
We consider the Newton stratification on Iwahori double cosets in the loop
group of a reductive group. We describe a group-theoretic condition on the
generic Newton point, called cordiality, under which the Newton poset (i.e. the
index set for non-empty Newton strata) is saturated and Grothendieck's
conjecture on closures of the Newton strata holds. Finally, we give several
large classes of Iwahori double cosets for which this condition is satisfied by
studying certain paths in the associated quantum Bruhat graph.Comment: 17 pages, 1 figure; expanded introduction, generalized main theorem,
changed section numbers; final version to appear in Forum of Mathematics,
Sigm
Conjugacy classes of finite groups and graph regularity
Given a finite group , denote by the simple undirected graph
whose vertices are the distinct sizes of noncentral conjugacy classes of ,
and set two vertices of to be adjacent if and only if they are not
coprime numbers. In this note we prove that, if is a -regular
graph with , then is a complete graph with vertices
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