arXiv subjects
Publications and source records attributed to Evgeny Vdovin.
Suppose that a finite solvable group $G$ acts faithfully, irreducibly and quasi-primitively on a finite vector space $V$. Then $G$ has a uniquely determined normal subgroup $E$ which is a direct product of extraspecial $p$-groups for various $p$ and we denote $e=\sqrt{|E/\bZ(E)|}$. We prove that when $e=2,3,4,8,9,16$, $G$ will have regular orbits on $V$ when the corresponding vector space is not too small.
In the paper we consider images of finite simple projective special linear and unitary groups under power words. In particular, we show that if $G\simeq \PSL_n^\varepsilon (q)$, then for every power words of type $x^M$ there exist constant $c$ and $N$ such that $\vert ω(G)\vert >c\frac{\vert G\vert }{n}$ whenether $\vert G\vert >N$.
Denote by $ν_p(G)$ the number of Sylow $p$-subgroups of $G$. It is not difficult to see that $ν_p(H)\leqν_p(G)$ for $H\leq G$, however $ν_p(H)$ does not divide $ν_p(G)$ in general. In this paper we reduce the question whether $ν_p(H)$ divides $ν_p(G)$ for every $H\leq G$ to almost simple groups. This result substantially generalizes the previous result by G. Navarro and also provides an alternative proof for the Navarro theorem.
Let $π$ be a set of primes. By H.Wielandt definition, {\it Sylow $π$-theorem} holds for a finite group $G$ if all maximal $π$-subgroups of $G$ are conjugate. In the paper, the following statement is proven. Assume that $π$ is a union of disjoint subsets $σ$ and $τ$ and a finite group $G$ possesses a $π$-Hall subgroup which is a direct product of a $σ$-subgroup and a $τ$-subgroup. Furthermore, assume that both the Sylow $σ$-theorem and $τ$-theorem hold for $G$. Then the Sylow $π$-theorem holds for $G$. This result confirms a conjecture posed by H.\,Wielandt in~1959.
In the paper, it is proved that if a finite group $G$ possesses a $π$-Hall subgroup for a set $π$ of primes, then every normal subgroup $A$ of $G$ possesses a $π$-Hall subgroup $H$ such that ${G=AN_G(H)}$.
N-Higgs doublet models (NHDM) are a popular framework to construct electroweak symmetry breaking mechanisms beyond the Standard model. Usually, one builds an NHDM scalar sector which is invariant under a certain symmetry group. Although several such groups have been used, no general analysis of symmetries possible in the NHDM scalar sector exists. Here, we make the first step towards this goal by classifying the elementary building blocks, namely the abelian symmetry groups, with a special emphasis on finite groups. We describe a strategy that identifies all abelian groups which are realizable as symmetry groups of the NHDM Higgs potential. We consider both the groups of Higgs-family transformations only and the groups which also contain generalized CP transformations. We illustrate this strategy with the examples of 3HDM and 4HDM and prove several statements for arbitrary N.