Related papers: On the intersections of solvable Hall subgroups in…

The following result is received: Let $H$ be a non-normal maximal subgroup of a finite solvable group $G$ and let $q \in \pi(F(H/\mathrm{Core}_GH))$, then $G$ has a Sylow $q$-subgroup $Q$ such that $N_{G}(Q) \subseteq H$.

Let $\sigma =\{\sigma_{i} | i\in I\}$ be a partition of the set of all primes $\Bbb{P}$ and $G$ a finite group. Let $\sigma (G)=\{\sigma _{i} : \sigma _{i}\cap \pi (G)\ne \emptyset$. A set ${\cal H}$ of subgroups of $G$ is said to be a…

Let G be a finite group. Denote by \psi(G) the sum \psi(G)=\sum_{x\in G}|x| where |x| denotes the order of the element x, and by o(G) the quotient o(G)=\frac{\psi(G)}{|G|}. Confirming a conjecture posed by E.I. Khukhro, A. Moreto and M.…

Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies the partial $ \Pi $-property in $ G $ if if there exists a chief series $ \varGamma_{G}: 1 =G_{0} < G_{1} < \cdot\cdot\cdot < G_{n}= G $ of $ G $ such that for…

This paper deals with combinatorial aspects of finite covers of groups by cosets or subgroups. Let $a_1G_1,...,a_kG_k$ be left cosets in a group $G$ such that ${a_iG_i}_{i=1}^k$ covers each element of $G$ at least $m$ times but none of its…

Let $G$ be a finite group and $\psi(G)=\sum_{g\in{G}}{o(g)}$. There are some results about the relation between $\psi(G)$ and the structure of $G$. For instance, it is proved that if $G$ is a group of order $n$ and…

Let $G$ be a finite group and $H$ be a subgroup of $G$. Then $H$ is said to be a $p$-$CAP$-subgroup of $G$, if $H$ covers or avoids any $pd$-chief factor of $G$. Furthermore, $H$ is said to be a strong $p$-$CAP$-subgroup of $G$, if for any…

Let \pi(G) denote the set of prime divisors of the order of a finite group G. The prime graph of G is the graph with vertex set \pi(G) with edges {p,q} if and only if there exists an element of order pq in G. In this paper, we prove that a…

We say that a subgroup $H$ is isolated in a group $G$ if for every $x\in G$ we have either $x\in H$ or $\langle x\rangle\cap H=1$. In this short note, we describe the set of isolated subgroups of a finite abelian group. The technique used…

Let $\mathbb{G}$ be a Lie group with solvable connected component and finitely-generated component group and $\alpha\in H^2(\mathbb{G},\mathbb{S}^1)$ a cohomology class. We prove that if $(\mathbb{G},\alpha)$ is of type I then the same…

It is shown that a closed solvable subgroup of a connected Lie group is compactly generated. In particular, every discrete solvable subgroup of a connected Lie group is finitely generated. Generalizations to locally compact groups are…

We consider the following problem stated by Vdovin (2010) in the "Kourovka notebook" (Problem 17.41): Let $H$ be a solvable subgroup of a finite group $G$ that has no nontrivial solvable normal subgroups. Do there always exist five…

Let G be a finite group and cd(G) denote the set of complex irreducible character degrees of G. In this paper, we prove that if G is a finite group and H is an almost simple group whose socle is a sporadic simple group H0 such that cd(G) =…

Let $\pi$ be a set of primes. We say that a finite group $G$ is a $\mathcal{D}_\pi$-group if the maximal $\pi$-subgroups of $G$ are conjugate. In this paper, we give an affirmative answer to Problem 17.44(b) from "Kourovka notebook", namely…

Let ${\mathbb M}$ be an arbitrary o-minimal structure. Let $G$ be a definably compact definably connected abelian definable group of dimension $n$. Here we compute the new the intrinsic o-minimal fundamental group of $G;$ for each $k>0$,…

Let G be a finite group. A subgroup M of G is said to be an NR-subgroup if, whenever K is normal in M, then K^G\cap M=K, where K^G is the normal closure of K in G. Using the Classification of Finite Simple Groups, we prove that if every…

We study the relationship between the existence of Hall $\pi$-subgroups and that of irreducible characters of $\pi'$-degree with prescribed fields of values in finite groups. This work extends a result of Navarro and Tiep from a single odd…

In this note we provide some counterexamples for the conjectures of finite simple groups, one of the conjectures said "all finite simple groups $G$ can be determined using their orders $|G|$ and the number of elements of order $p$, where…

We introduce the notion of a probabilistic identity of a residually finite group. We prove that a finitely generated linear group satisfies a probabilistic identity if and only if it is virtually solvable. As an application, we prove a…

By a proper cover of a finite group G we mean an extension of a nontrivial finite group by G. Our purpose is to show that a proper cover of a finite simple group L of Lie type always contains an element whose order differs from the element…