UDC 512.542Keywords: finite group, finite simple group, group of Lie type, spectrum of a finite group, recognition by spectrum, prime graph of a finite group, independence number of a prime graph, 2-independence number of a prime graph.For every finite non-Abelian simple group, we give an exhaustive arithmetic criterion for adjacency of vertices in a prime graph of the group. For the prime graph of every finite simple group, this criterion is used to determine an independent set with a maximal number of vertices and an independent set with a maximal number of vertices containing 2, and to define orders on these sets; the information obtained is collected in tables. We consider several applications of these results to various problems in finite group theory, in particular, to the recognition-by-spectra problem for finite groups.

We report the recent progress in understanding of symmetries which can be implemented in the scalar sector of electroweak symmetry breaking models with several Higgs doublets. In particular we present the list of finite reparametrization symmetry groups which can appear in the three-Higgs-doublet models.

Symmetries play a crucial role in electroweak symmetry breaking models with nonminimal Higgs content. Within each class of these models, it is desirable to know which symmetry groups can be implemented via the scalar sector. In N -Higgs-doublet models, this classification problem was solved only for N = 2 doublets. Very recently, we suggested a method to classify all realizable finite symmetry groups of Higgs-family transformations in the three-Higgs-doublet model (3HDM). Here, we present this classification in all detail together with an introduction to the theory of solvable groups, which play the key role in our derivation. We also consider generalized-CP symmetries, and discuss the interplay between Higgs-family symmetries and CP -conservation. In particular, we prove that presence of the Z 4 symmetry guarantees the explicit CP -conservation of the potential. This work completes classification of finite reparametrization symmetry groups in 3HDM.

N-Higgs-doublet models (NHDM) are among the most popular examples of electroweak symmetry breaking mechanisms beyond the Standard Model. Discrete symmetries imposed on the NHDM scalar potential play a pivotal role in shaping the phenomenology of the model, and various symmetry groups have been studied so far. However, in spite of all efforts, the classification of finite Higgs-family symmetry groups realizable in NHDM for any N>2 is still missing. Here, we solve this problem for the three-Higgs-doublet model by making use of Burnside's theorem and other results from pure finite group theory which are rarely exploited in physics. Our method and results can be also used beyond high-energy physics, for example, in study of possible symmetries in three-band superconductors.Comment: 5 pages; v2: expanded introduction, some minor corrections, matches the published versio

A prime graph of a finite group is defined in the following way: the set of vertices of the graph is the set of prime divisors of the group, and two distinct vertices r and s are adjacent, if there is an element of order rs in the group. In this paper we continue our investigation of the prime graph of a finite simple group started in [1], namely we describe all cocliques of maximal size for all finite simple groups.