Related papers: Large subgroups in finite groups

We give a detailed description of infinite locally nilpotent groups G such that the index |C_G (x) : <x>| is finite, for every non-normal cyclic subgroup <x> of G. We are also able to extend our analysis to all non-periodic groups…

We study adequate subgroups of $GL_n$ over a finite field. This notion is useful in the study of automorphy lifting theorems. In particular, we give a sufficient condition for a subgroup to be adequate.

We give a characterization of the finite groups having nilpotent or abelian Hall $\pi$-subgroups which can easily be verified from the character table.

We prove that the product of a subset and a normal subset inside any finite simple non-abelian group $G$ grows rapidly. More precisely, if $A$ and $B$ are two subsets with $B$ normal and neither of them is too large inside $G$, then $|AB|…

The existence of invariant transversals for a normal subgroup $H$ in a group $G$ is investigated. This yields counterexamples to a conjecture in case $H$ is abelian and $G$ is finite.

A proper subgroup $H$ in a finite group $G$ is said to be large if $|H|^3\geq |G|$. In this paper, we determined all large maximal subgroups of almost simple classical groups. Combined with the work of Alavi and Burness (J. Algebra 421…

Let $G$ be a finite group and $\sigma =\{\sigma_{i} | i\in I\}$ some partition of the set of all primes $\Bbb{P}$, that is, $\sigma =\{\sigma_{i} | i\in I \}$, where $\Bbb{P}=\bigcup_{i\in I} \sigma_{i}$ and $\sigma_{i}\cap \sigma_{j}=…

Every finite group $G$ has a normal series each of whose factors is either a solvable group or a direct product of nonabelian simple groups. The minimum number of nonsolvable factors attained on all possible such series is called the…

We introduce the notion of the depth of a finite group $G$, defined as the minimal length of an unrefinable chain of subgroups from $G$ to the trivial subgroup. In this paper we investigate the depth of (non-abelian) finite simple groups.…

We use recurrence relations to derive explicit formulas for counting the number of subgroups of given order (or index) in rank 3 finite abelian p-groups and use these to derive similar formulas in few cases for rank 4. As a consequence, we…

We show that if a group G is finitely presented and nilpotent-by-abelian-by-finite, then there is an upper bound on the first betti number of M as M runs through all subgroups of finite index in G.

This paper is a survey on the works [MS77, MS79, MS81] on maximal subgroups in finitely generated linear groups, and the works that followed it [GG08, GG13b, GG13a, Kap03, Iva92, HO16, GM16, AGS14, Sf90, Sf98, Per05, AKT16, FG18, GS17]…

We give a description of a finite group whose maximal subgroups possess only soluble proper subgroups, which implies the answer to the well-known question on composition factors of finite groups, whose second maximal subgroups are soluble.

Suppose that $G$ is a finite group and $H$ is a nilpotent subgroup of $G$. If a character of $H$ induces an irreducible character of $G$, then the generalized Fitting subgroup of $G$ is nilpotent.

An automorphism $\alpha$ of a group $G$ is normal if it fixes every normal subgroup of $G$ setwise. We give an algebraic description of normal automorphisms of relatively hyperbolic groups. In particular, we prove that for any relatively…

A finite group $G$ is called a Schur group if every $S$-ring over $G$ is schurian, i.e. associated in a natural way with a subgroup of $\sym(G)$ that contains all right translations. We prove that every nonabelian nilpotent Schur group…

Given a finite group $G$, we denote by $\nu(G)$ the probability that two randomly chosen elements of $G$ generate a nilpotent subgroup. We prove that if $\nu(G)>1/12,$ then $G$ is solvable.

Let $G$ be a finite group and $N_{\Omega}(G)$ be the intersection of the normalizers of all subgroups belonging to the set $\Omega(G),$ where $\Omega(G)$ is a set of all subgroups of $G$ which have some theoretical group property. In this…

A finite group $G$ is called uniformly semi-rational if there exists an integer $r$ such that the generators of every cyclic sugroup $\langle x \rangle$ of $G$ lie in at most two conjugacy classes, namely $x^G$ or $(x^r)^G$. In this paper,…

Let $G$ be a finite group and let $H$ be a subgroup of $G$. We say that $H$ is extremely closed in $G$ if $\langle H,H^g\rangle\cap N_G(H)=H$ for all $g\in G.$ In this paper, we determine the structure of finite groups with an extremely…