On a Finite Group Generated by Subnormal Supersoluble Subgroups

On finite groups whose Sylow subgroups are submodular

A subgroup $H$ of a finite group $G$ is called submodular in $G$, if we can connect $H$ with $G$ by a chain of subgroups, each of which is modular (in the sense of Kurosh) in the next. If a group $G$ is supersoluble and every Sylow subgroup of $G$ is submodular in $G$, then $G$ is called strongly supersoluble. The properties of groups with submodular Sylow subgroups are obtained. In particular, we proved that in a group every Sylow subgroup is submodular if and only if the group is Ore dispersive and every its biprimary subgroup is strongly supersoluble.

A note on the supersolvability of a finite group with prime index of some subgroups

In this paper, we proved that a group $G$ is supersoluble if and only if for any prime $p\in π(G)$ there exists a supersoluble subgroup of index $p$.

超可解补群关于阿贝尔基底的共轭条件及非互素作用的不动点结果

Conjugacy conditions for supersoluble complements of an abelian base and a fixed point result for non-coprime actions

本文研究了有限分裂扩张中阿贝尔基底的超可解补群的共轭条件，证明了基于 Sylow $p$-子群共轭性的充要条件，并放宽了 Higman 的限制，最后得到了一个类似 Glauberman 引理的非互素作用不动点结果。

Finite groups with all 3-maximal subgroups K-U-subnormal

A classification of finite groups in which every 3-maximal subgroup is K-U-subnormal is given.

On grouops with formational subnormal or self-normalizing subgroups

We establish the structure of finite groups with $\mathfrak{F}$-subnormal or self-normalizing primary cyclic subgroups in case $\mathfrak{F}$ is a subgroup-closed saturated superradical formation containing all nilpotent groups.

Finite groups in which every self-centralizing subgroup is a TI-subgroup or subnormal or has $p'$-order

We first give complete characterizations of the structure of finite group $G$ in which every subgroup (or non-nilpotent subgroup, or non-abelian subgroup) is a TI-subgroup or subnormal or has $p'$-order for a fixed prime divisor $p$ of $|G|$. Furthermore, we prove that every self-centralizing subgroup (or non-nilpotent subgroup, or non-abelian subgroup) of $G$ is a TI-subgroup or subnormal or has $p'$-order for a fixed prime divisor $p$ of $|G|$ if and only if every subgroup (or non-nilpotent subgroup, or non-abelian subgroup) of $G$ is a TI-subgroup or subnormal or has $p'$-order. Based on th…

On groups in which subnormal subgroups of infinite rank are commensurable with some normal subgroup

We study soluble groups G in which each subnormal subgroup H with infinite rank is commensurable with a normal subgroup, i.e. there exists a normal subgroup N such that the intersection of H and N has finite index in both H and N. We show that if such a G is periodic, then all subnormal subgroups are commensurable with a normal subgroup, provided either the Hirsch-Plotkin radical of G has infinite rank or G is nilpotent-by-abelian (and has infinite rank).

Finite groups with $\mathbb P$-subnormal primary cyclic subgroups

A subgroup $H$ of a group $G$ is called $\mathbb P$-subnormal in $G$ whenever either $H=G$ or there is a chain of subgroups $H=H_0\subset H_1\subset ... \subset H_n=G$ such that $|H_i:H_{i-1}|$ is a prime for all $i$. In this paper, we study the groups in which all primary cyclic subgroups are $\mathbb P$-subnormal.

Finite groups with $\Bbb P$-subnormal 2-maximal subgroups

A subgroup $H$ of a group $G$ is called $\Bbb P$-{\sl subnormal} in $G$ if either $H=G$ or there is a chain of subgroups $H=H_0\subset H_1\subset...\subset H_n=G$ such that $|H_i:H_{i-1}|$ is prime for $1\le i\le n$. In this paper we study the groups all of whose 2-maximal subgroups are $\Bbb P$-subnormal.

On groups whose subnormal subgroups are inert

A subgroup H of a group G is called inert if for each $g\in G$ the index of $H\cap H^g$ in $H$ is finite. We give a classification of soluble-by-finite groups $G$ in which subnormal subgroups are inert in the cases where $G$ has no nontrivial torsion normal subgroups or $G$ is finitely generated.

On groups whose subnormal abelian subgroups are normal

In the current paper we study the groups, whose subnormal abelian subgroups are normal. We obtained a quite detailed description of such hyperabelian groups with a periodic Baer radical. The description of hyperabelian Lie algebras, whose abelian subideals are ideals, is also obtained.

Subnormal subgroups in free groups, their growth and cogrowth

In this paper, the author (1) compares subnormal closures of finite sets in free groups; (2) proves that the exponential growth rate (e.g.r.), i.e., the limit of the n-th roots of g(n), where g(n) is the growth function of a subgroup H with respect to a finite free basis of F, exists for any subgroup H of the free group F; (3) gives sharp estimates from below for the e.g.r. of subnormal subgroups in free groups; and (4) finds cogrowth for the subnormal closures of free generators in F.

弱次正规子群与Baer-Suzuki定理的变体

Weakly subnormal subgroups and variations of the Baer-Suzuki theorem

该研究针对有限群中的弱次正规$p$-子群展开分类工作，进一步刻画含循环弱次正规$p$-子群的有限群，并借助群元素阶的性质推导Baer-Suzuki定理的多种变体，丰富有限群论相关结论。

$σ$-次正规子群的并

Joins of $σ$-subnormal subgroups

该研究针对局部有限群中$σ$-次正规子群的并不再保持$σ$-次正规性的问题，探究相关群类的扩展边界，给出两个$σ$-次正规子群的并为$σ$-次正规的充要条件，推广了Roseblade的结果。

Products of subgroups, subnormality, and relative orders of elements

Let $G$ be a group. We give an explicit description of the set of elements $x \in G$ such that $x^{|G:H|} \in H$ for every subgroup of finite index $H \leqslant G$. This is related to the following problem: given two subgroups $H$ and $K$, with $H$ of finite index, when does $|HK:H|$ divide $|G:H|$?

Subnormalizers and solvability in finite groups

For a finite group $G$, we study the probability $sp(G)$ that, given two elements $x,y \in G$, the cyclic subgroup $\langle x \rangle$ is subnormal in the subgroup $\langle x, y \rangle$. This can be seen as an intermediate invariant between the probability that two elements generate a nilpotent subgroup and the probability that two elements generate a solvable subgroup. We prove that $sp(G) \leq 1/6$ for every nonsolvable group $G$.

Finite groups with systems of $K$-$\frak{F}$-subnormal subgroups

Let $\frak {F}$ be a class of group. A subgroup $A$ of a finite group $G$ is said to be $K$-$\mathfrak{F}$-subnormal in $G$ if there is a subgroup chain $$A=A_{0} \leq A_{1} \leq \cdots \leq A_{n}=G$$ such that either $A_{i-1} \trianglelefteq A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}} \in \mathfrak{F}$ for all $i=1, \ldots , n$. A formation $\frak {F}$ is said to be $K$-lattice provided in every finite group $G$ the set of all its $K$-$\mathfrak{F}$-subnormal subgroups forms a sublattice of the lattice of all subgroups of $G$. In this paper we consider some new applications of the theory of $K$-lattic…

On groups with all subgroups subnormal or soluble of bounded derived length

In this paper, we deal with locally graded groups whose subgroups are either subnormal or soluble of bounded derived length, say d. In particular, we prove that every locally (soluble-by-finite) group with this property is either soluble or an extension of a soluble group of derived length at most d by a finite group, which fits between a minimal simple group and its automorphism group. We also classify all the finite non-abelian simple groups whose proper subgroups are metabelian.

Finite groups with $\mathfrak{F}$-subnormal normalizers of Sylow subgroups

Let $π$ be a set of primes and $\mathfrak{F}$ be a formation. In this article a properties of the class ${\rm w}^{*}_π\mathfrak{F}$ of all groups $G$, such that $π(G)\subseteq π(\mathfrak{F})$ and the normalizers of all Sylow $p$-subgroups of $G$ are $\mathfrak{F}$-subnormal in $G$ for every $p\inπ\capπ(G)$ are investigated. It is established that ${\rm w}^{*}_π\mathfrak{F}$ is a formation. Some hereditary saturated formations $\mathfrak{F}$ for which ${\rm w}^{*}_π\mathfrak{F}=\mathfrak{F}$ are founded.

所有子群均为亏格至多5的次正规的无扭幂零类6群

A torsion-free group of nilpotency class six with all subgroups subnormal of defect at most $5$

该研究构造了一个2生成的无扭幂零群，其幂零类为6且所有子群均为亏格至多5的次正规，从而否定了Casolo关于此类群幂零类与亏格关系的问题在n=5时的结论。

有限群的$\boldsymbol{\rm K}$-$\boldsymbol{\bb P}_{t}$-次正规子群及相关群系

On $\mathrm{K}$-$\mathbb{P}_{t}$-subnormal subgroups of finite groups and related formations

本文定义了有限群中的${\rm K}$-$\bb P_{t}$-次正规子群概念，研究了这类子群的性质，并刻画了所有Sylow子群均为${\rm K}$-$\bb P_{t}$-次正规的有限群类的相关性质，丰富了有限群子群结构与群系理论的研究内容。

A generalisation of a theorem of Wielandt

In 1974, Helmut Wielandt proved that in a finite group $G$, a subgroup $A$ is subnormal if and only if it is subnormal in every $\seq{A,g}$ for all $g\in G$. In this paper, we prove that the subnormality of an odd order nilpotent subgroup $A$ of $G$ is already guaranteed by a seemingly weaker condition: $A$ is subnormal in $G$ if for every conjugacy class $C$ of $G$ there exists $c\in C$ for which $A$ is subnormal in $\seq{A,c}$. We also prove the following property of finite non-abelian simple groups: if $A$ is a subgroup of odd prime order $p$ in a finite almost simple group $G$, then there …

Finite groups whose $n$-maximal subgroups are $σ$-subnormal

Let $σ=\{σ_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$. A set ${\cal H}$ of subgroups of $G$ is said to be a \emph{complete Hall $σ$-set} of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $σ_{i}$-subgroup of $G$, for some $i\in I$, and $\cal H$ contains exact one Hall $σ_{i}$-subgroup of $G$ for every $σ_{i}\in σ(G)$. A subgroup $H$ of $G$ is said to be: \emph{$σ$-permutable} or \emph{$σ$-quasinormal} in $G$ if $G$ possesses a complete Hall $σ$-set set ${\cal H}$ such that $HA^{x}=A^{x}H$ for all $A\in {\cal H}$ and $x\in G$: \emph{$σ$-subnormal} in $G$ if there i…

Generalized Fitting subgroups of finite groups

In this paper we consider the Fitting subgroup $F(G)$ of a finite group $G$ and its generalizations: the quasinilpotent radical $F^*(G)$ and the generalized Fitting subgroup $\tilde{F}(G)$ defined by $\tilde{F}(G)\supseteq Φ(G)$ and $\tilde{F}(G)/Φ(G)=Soc(G/Φ(G))$. We sum up known properties of $\tilde{F}(G)$ and suggest some new ones. Let $R$ be a subgroup of a group $G$. We shall call a subgroup $H$ of $G$ the $R$-subnormal subgroup if $H$ is subnormal in $ \langle H,R\rangle$. In this work the influence of $R$-subnormal subgroups (maximal, Sylow, cyclic primary) on the structure of finite g…

On subnormal subgroups in division rings containing a non-abelian solvable subgroup

Let $D$ be a division ring with center $F$ and $N$ a subnormal subgroup of the multiplicative group $D^*$ of $D$. Assume that $N$ contains a non-abelian solvable subgroup. In this paper, we study the problem on the existence of non-abelian free subgroups in $N$. In particular, we show that if either $N$ is algebraic over $F$ or $F$ is uncountable, then $N$ contains a non-abelian free subgroup.

On some applications of Fitting like subgroups of finite groups

In this paper we study the groups all whose maximal or all Sylow subgroups are $K$-$\mathfrak{F}$-subnormal in their product the with generalizations of the Fitting subgroup $\mathrm{F}^*(G)$ and $\mathrm{\tilde F}(G)$. We prove that a hereditary formation $\mathfrak{F}$ contains every group all whose Sylow subgroups are $K$-$\mathfrak{F}$-subnormal in their product with $\mathrm{F}^*(G)$ if and only if $\mathfrak{F}$ is the class of all $σ$-nilpotent groups for some partition $σ$ of the set of all primes. We obtain a new characterization of the $σ$-nilpotent hypercenter, i.e. the $\mathfrak{F…

On $H_σ$-permutably embedded and $H_σ$-subnormaly embedded subgroups of finite groups

Let $G$ be a finite group. Let $σ=\{σ_{i} | i\in I\}$ be a partition of the set of all primes $\Bbb{P}$ and $n$ an integer. We write $σ(n) =\{σ_{i} |σ_{i}\cap π(n)\ne \emptyset \}$, $σ(G) =σ(|G|)$. A set $ {\cal H}$ of subgroups of $G$ is said to be a complete Hall $σ$-set of $G$ if every member of ${\cal H}\setminus \{1\}$ is a Hall $σ_{i}$-subgroup of $G$ for some $σ_{i}$ and ${\cal H}$ contains exact one Hall $σ_{i}$-subgroup of $G$ for every $σ_{i}\in σ(G)$. A subgroup $A$ of $G$ is called: (i) a $σ$-Hall subgroup of $G$ if $σ(|A|) \cap σ(|G:A|)=\emptyset$; (ii) $σ$-permutable in $G$ if $G…

论有限群中的 $\mathrm{K}\mathfrak F$-次正规性与子模性

On $\mathrm{K}\mathfrak F$-subnormality and submodularity in a finite group

本文研究有限群中素子群的子模性与 $\mathrm{K}\mathfrak U_1$-次正规性的等价关系，并证明可解 $\mathrm{K}\mathfrak{F}$-次正规子群必含于可解根。

关于有限群σ-性质理论的一个贡献

A contribution to the theory of $σ$-properties of a finite group

本文通过引入正规子群D避开主因子的条件，分别给出了有限群超可解性与有限可解群为PST-群的充要刻画，推进了有限群σ-性质理论。

2-subnormal quadratic offenders and Oliver's p-group conjecture

Bob Oliver conjectures that if $p$ is an odd prime and $S$ is a finite $p$-group, then the Oliver subgroup $\X(S)$ contains the Thompson subgroup $J_e(S)$. A positive resolution of this conjecture would give the existence and uniqueness of centric linking systems for fusion systems at odd primes. Using ideas and work of Glauberman, we prove that if $p \geq 5$, $G$ is a finite $p$-group, and $V$ is an elementary abelian $p$-group which is an F-module for $G$, then there exists a quadratic offender which is 2-subnormal (normal in its normal closure) in $G$. We apply this to show that Oliver's co…

The continuation of finite $E_\mathfrak{F}$-groups theory

We describe the structure of finite groups with $\mathfrak{F}$-subnormal or self-normalizing primary cyclic subgroups when $\mathfrak{F}$ is a subgroup-closed saturate superradical formation containing all nilpotent groups. We prove that groups with absolutely $\mathfrak{F}$-subnormal or self-normalizing primary cyclic subgroups are soluble when $\mathfrak{F}$ is a subgroup-closed saturate formation containing all nilpotent groups.

Formations of Finite Groups in Polynomial Time: $\mathfrak{F}$-residuals and $\mathfrak{F}$-subnormality

For a wide family of formations $\mathfrak{F}$ it is proved that the $ \mathfrak{F}$-residual of a permutation finite group can be computed in a polynomial time. Moreover, if in the previous case $\mathfrak{F}$ is hereditary, then an $\mathfrak{F}$-subnormality of a subgroup can be checked in a polynomial time.

On factorized groups with permutable subgroups of factors

The subgroups $A$ and $B$ of a group~$G$ are called {\rm msp}-permutable, if the following statements hold: $AB$~is a subgroup of~$G$; the subgroups $P$ and $Q$ are mutually permutable, where $P$~is an arbitrary Sylow $p$-subgroup of~$A$ and $Q$~is an arbitrary Sylow $q$-subgroup of~$B$, ${p\neq q}$. In the present paper, we investigate groups that factorized by two {\rm msp}-permutable subgroups. In particular, the supersolubility of the product of two supersoluble {\rm msp}-permutable subgroups is proved.

On $n$-maximal subgroups of finite groups

We describe finite soluble groups in which every $n$-maximal subgroup is $\mathfrak F$-subnormal.

A generalisation of Schenkman's theorem

Let $G$ be a finite group and let $\mathfrak{F}$ be a hereditary saturated formation. We denote by $\mathbf{Z}_{\mathfrak{F}}(G)$ the product of all normal subgroups $N$ of $G$ such that every chief factor $H/K$ of $G$ below $N$ is $\mathfrak{F}$-central in $G$, that is, \[ (H/K) \rtimes (G/\mathbf{C}_{G}(H/K)) \in \mathfrak{F}. \]A subgroup $A \leq G$ is said to be $\mathfrak{F}$-subnormal in the sense of Kegel, or $K$-$\mathfrak{F}$-subnormal in $G$, if there is a subgroup chain \[ A = A_0 \leq A_1 \leq \ldots \leq A_n = G \] such that either $A_{i-1} \trianglelefteq A_{i}$ or $A_i / (A_{i-1…

A generalization of 2-Baer groups

A group in which all cyclic subgroups are 2-subnormal is called a 2-Baer group. The topic of this paper are generalized 2-Baer groups, i.e. groups in which the non-2-subnormal cyclic subgroups generate a proper subgroup of the group. If this subgroup is non-trivial, the group is called a generalized T_2-group. In particular, we provide structure results for such groups, investigate their nilpotency class and construct examples of finite p-groups which are generalized T_2-groups.

Algebraic commutators with respect to subnormal subgroups in division rings

Let $D$ be a division ring and $K$ a subfield of $D$ which is not necessarily contained in the center $F$ of $D$. In this paper, we study the structure of $D$ under the condition of left algebraicity of certain subsets of $D$ over $K$. Among results, it is proved that if $D^*$ contains a noncentral normal subgroup which is left algebraic over $K$ of bounded degree $d$, then $[D:F]\le d^2$. In case $K=F$, the obtained results show that if either all additive commutators or all multiplicative commutators with respect to a noncentral subnormal subgroup of $D^*$ are algebraic of bounded degree $d$…

On a generalization of one Kramer's theorem

Yangming Li and Xianhua Li in 2012 proposed a conjecture that generalizes O.U. Kramer's result about supersoluble groups. Here we proved that this conjecture is false in the general case and true for groups with the trivial Frattini subgroup.

On one generalization of finite $\frak U$-critical groups

A proper subgroup $H$ of a group $G$ is said to be: $\Bbb{P}$-subnormal in $G$ if there exists a chain of subgroups $H=H_0 < H_1< ... < H_{n}=G$ such that $|H_{i}:H_{i-1}|$ is a prime for $i=1,...,n$; $\Bbb{P}$-abnormal in $G$ if for every two subgroups $K\leq L$ of $G$, where $H\leq K$, $|L:K|$ is not a prime. In this paper we describe finite groups in which every non-identity subgroup is either $\Bbb{P}$-subnormal or $\Bbb{P}$-abnormal.

次正规化子猜想与幺模特征

The subnormaliser conjecture and unipotent characters

本文通过推导李型近单群的幺模特征的泛型双射，证明了有限群特征双射的次正规化子猜想的若干实例。

The permutability of $σ_i$-sylowizers of some $σ_i$-subgroups in finite groups

Let $σ=\{σ_{i}|i\in I\}$ be a partition of the set of all primes $\mathbb{P}$, $G$ a finite group and $σ(G)=\{σ_{i}|σ_{i}\cap π(|G|)\neq\emptyset\}$. A subgroup $S$ of a group $G$ is called a $σ_i$-sylowizer of a $σ_i$-subgroup $R$ in $G$ if $S$ is maximal in $G$ with respect to having $R$ as its Hall $σ_i$-subgroup. The main aim of this paper is to investigate the influence of $σ_i$-sylowizers on the structure of finite groups. We obtained some new characterizations of supersoluble groups by the permutability of the $σ_i$-sylowizers of some $σ_i$-subgroups.

On generalizations of Baer's theorems about the hypercenter of a finite group

We investigate the intersection of normalizers and $\mathfrak{F}$-subnormalizers of different types of systems of subgroups ($\mathfrak{F}$-maximal, Sylow, cyclic primary). We described all formations $\mathfrak{F}=\underset{i\in I}\times\mathfrak{F}_{π_i}$ for which the intersection of normalizers of all $\mathfrak{F}_i$-maximal subgroups of $G$ is the $\mathfrak{F}$-hypercenter of $G$ for every group $G$. Also we described all formations $\mathfrak{F}$ for which the intersection of $\mathfrak{F}$-subnormalizers of all Sylow (cyclic primary) subgroups of $G$ is the $\mathfrak{F}$-hypercenter …

On algebras associated with invariant means on the subnormal subgroups of an amenable group

Let $G$ be an amenable group. We define and study an algebra $\mathcal{A}_{sn}(G)$, which is related to invariant means on the subnormal subgroups of $G$. For a just infinite amenable group $G$, we show that $\mathcal{A}_{sn}(G)$ is nilpotent if and only if $G$ is not a branch group, and in the case that it is nilpotent we determine the index of nilpotence. We next study $\operatorname{rad} \ell^1(G)^{**}$ for an amenable branch group $G$, and show that it always contains nilpotent left ideals of arbitrarily large index, as well as non-nilpotent elements. This provides infinitely many finitely…

关于具有模 Schmidt 子群的群

On groups with modular Schmidt subgroups

本文证明了若群 $G$ 的每个 Schmidt 子群次正规或模，则 $G/F(G)$ 循环。

Ingleton不等式在亚循环群中成立且在超可解群中不成立

The Ingleton inequality holds for metacyclic groups and fails for supersoluble groups

本文证明了Ingleton不等式在亚循环群中恒成立，但在超可解群中不成立，并验证了阶小于1024的幂零群无反例，同时提供了用于检查该不等式的Magma包。

有限群在多项式时间内的群系：$\mathfrak{F}$-超中心

Formations of Finite Groups in Polynomial Time: the $\mathfrak{F}$-Hypercenter

本文证明了一大类群系（含 Baer-局部群系）的 $\mathfrak{F}$-超中心可在多项式时间内计算，并给出了多种具体群类的算法及部分极大子群交的计算方法。

On a generalisation of finite $T$-groups

Let $σ=\{σ_i |i\in I\}$ is some partition of all primes $\mathbb{P}$ and $G$ a finite group. A subgroup $H$ of $G$ is said to be $σ$-subnormal in $G$ if there exists a subgroup chain $H=H_0\leq H_1\leq \cdots \leq H_n=G$ such that either $H_{i-1}$ is normal in $H_i$ or $H_i/(H_{i-1})_{H_i}$ is a finite $σ_j$-group for some $j \in I$ for $i = 1, \ldots, n$. We call a finite group $G$ a $T_σ$-group if every $σ$-subnormal subgroup is normal in $G$. In this paper, we analyse the structure of the $T_σ$-groups and give some characterisations of the $T_σ$-groups.

Products of locally cyclic groups

We consider groups of the form $G=AB$ with two locally cyclic subgroups $A$ and $B$. The structure of these groups is determined in the cases when $A$ and $B$ are both periodic or when one of them is periodic and the other is not. Together with a previous study of the case when $A$ and $B$ are torsion-free, this gives a complete classification of all groups that are the product of two locally cyclic groups. It is also shown that the product of a finite number of pairwise permutable periodic locally cyclic subgroups is a locally supersoluble group.

关于融合系统和局部性的威拉德定理

On Wielandt's Join Theorem for fusion systems and localities

本文研究融合系统和局部性中的威拉德定理，证明了其在融合系统和局部性中的版本，并提出一个独立的群论结果。

The generalised Fitting subgroup of a profinite group

The generalised Fitting subgroup of a finite group is the group generated by all subnormal subgroups that are either nilpotent or quasisimple. The importance of this subgroup in finite group theory stems from the fact that it always contains its own centraliser, so that any finite group is an abelian extension of a group of automorphisms of its generalised Fitting subgroup. We define a class of profinite groups which generalises this phenomenon, and explore some consequences for the structure of profinite groups.

关于有限群 $ p $-超可解性的一个注记

A note on $ p $-supersolubility of finite groups

本文通过引入子群满足 $ \mathscr L $-$ Π$-性质的假设条件，给出了有限 $ p $-超可解群的一个刻画。

On the structure of some locally nilpotent groups without contranormal subgroups

Following J.S. Rose, a subgroup H of a group G is said contranormal in G if G = H^G . In a certain sense, contranormal subgroups are antipodes to subnormal subgroups. It is well known that a finite group is nilpotent if and only if it has no proper contranormal subgroups. We prove that a nilpotent-by-finite group with no proper contranormal subgroup is nilpotent. There are locally nilpotent groups with a proper contranormal subgroup. We study the structure of hypercentral groups with a finite proper contranormal subgroup.

On the Frattini subgroup of a finite group

We study the class of finite groups $G$ satisfying $Φ(G/N)= Φ(G)N/N$ for all normal subgroups $N$ of $G$. As a consequence of our main results we extend and amplify a theorem of Doerk concerning this class from the soluble universe to all finite groups and answer in the affirmative a long-standing question of Christensen whether the class of finite groups which possess complements for each of their normal subgroups is subnormally closed.

Algorithms for arithmetic groups with the congruence subgroup property

We develop practical techniques to compute with arithmetic groups $H\leq \mathrm{SL}(n,\mathbb{Q})$ for $n>2$. Our approach relies on constructing a principal congruence subgroup in $H$. Problems solved include testing membership in $H$, analyzing the subnormal structure of $H$, and the orbit-stabilizer problem for $H$. Effective computation with subgroups of $\mathrm{GL}(n,\mathbb{Z}_m)$ is vital to this work. All algorithms have been implemented in GAP.