10,054 research outputs found
Cocliques of maximal size in the prime graph of a finite simple group
In this paper we continue our investgation of the prime graph of a finite
simple group started in http://arxiv.org/abs/math/0506294 (the printed version
appeared in [1]). We describe all cocliques of maximal size for all finite
simple groups and also we correct mistakes and misprints from our previous
paper. The list of correction is given in Appendix of the present paper.Comment: published version with correction
Groups with the same prime graph as the simple group Dn(5)
Let G be a finite group. The prime graph of G is denoted by Γ(G). Let G be a finite group such that Γ(G)=Γ(Dn(5)), where n≥6. In the paper, as the main result, we show that if n is odd, then G is recognizable by the prime graph and if n is even, then G is quasirecognizable by the prime graph.Нехай G — скшченна група. Простий граф групи GG позначимо через Γ(G). Нехай G — скінченна група така, що Γ(G)=Γ(Dn(5)), де n≥6. Як основний результат роботи доведено, що для непарних n група G розтзнається простим графом, а для парних n група G є такою, що квазiрозпiзнається простим графом
Distances and Isomorphism between Networks and the Stability of Network Invariants
We develop the theoretical foundations of a network distance that has
recently been applied to various subfields of topological data analysis, namely
persistent homology and hierarchical clustering. While this network distance
has previously appeared in the context of finite networks, we extend the
setting to that of compact networks. The main challenge in this new setting is
the lack of an easy notion of sampling from compact networks; we solve this
problem in the process of obtaining our results. The generality of our setting
means that we automatically establish results for exotic objects such as
directed metric spaces and Finsler manifolds. We identify readily computable
network invariants and establish their quantitative stability under this
network distance. We also discuss the computational complexity involved in
precisely computing this distance, and develop easily-computable lower bounds
by using the identified invariants. By constructing a wide range of explicit
examples, we show that these lower bounds are effective in distinguishing
between networks. Finally, we provide a simple algorithm that computes a lower
bound on the distance between two networks in polynomial time and illustrate
our metric and invariant constructions on a database of random networks and a
database of simulated hippocampal networks
Criterion of unrecognizability of a finite group by its Gruenberg-Kegel graph
The Gruenberg-Kegel graph associated with a finite group has
as vertices the prime divisors of , with an edge from to if and
only if contains an element of order . This graph has been the subject
of much recent interest; one of our goals here is to give a survey of some of
this material, relating to groups with the same Gruenberg-Kegel graph. However,
our main aim is to prove several new results. Among them are the following.
- There are infinitely many finite groups with the same Gruenberg-Kegel graph
as the Gruenberg-Kegel of a finite group if and only if there is a finite
group with non-trivial solvable radical such that .
- There is a function on the natural numbers with the property that if a
finite -vertex graph whose vertices are labelled by pairwise distinct primes
is the Gruenberg-Kegel graph of more than finite groups, then it is the
Gruenberg-Kegel graph of infinitely many finite groups. (The function we give
satisfies , but this is probably not best possible.)
- If a finite graph whose vertices are labelled by pairwise distinct
primes is the Gruenberg-Kegel graph of only finitely many finite groups, then
all such groups are almost simple; moreover, has at least three
pairwise non-adjacent vertices, and is non-adjacent to at least one odd
vertex.
- Groups whose power graphs, or commuting graphs, are isomorphic have the
same Gruenberg-Kegel graph.
- The groups and are uniquely determined by the
isomorphism types of their Gruenberg-Kegel graphs.
In addition, we consider groups whose Gruenberg-Kegel graph has no edges.
These are the groups in which every element has prime power order, and have
been studied under the name \emph{EPPO groups}; completing this line of
research, we give a complete list of such groups.Comment: 29 page
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