Marius Tarnauceanu
The main goal of this note is to determine and to count the normal subgroups of a ZM-group. We also indicate some necessary and sufficient conditions such that the normal subgroups of a ZM-group form a chain.
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We examine distortion of finitely generated normal subgroups. We show a connection between subgroup distortion and group divergence. We suggest a method computing the distortion of normal subgroups by decomposing the whole group into smaller subgroups. We apply our work to compute the distortion of normal subgroups of graph of groups and normal subgroups of right-angled Artin groups that induce infinite cyclic quotient groups. We construct normal subgroups of $\CAT(0)$ groups introduced by Macura and introduce a collection of normal subgroups of right-angled Artin groups. These groups provide …
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).
Among compact Hausdorff groups G whose maximal profinite quotient is finitely generated, we characterize those that possess a proper dense normal subgroup. We also prove that the abstract commutator subgroup [H,G] is closed for every closed normal subgroup H of G.
We inspect the normal subgroup structure of the braided Thompson groups Vbr and Fbr. We prove that every proper normal subgroup of Vbr lies in the kernel of the natural quotient Vbr \onto V, and we exhibit some families of interesting such normal subgroups. For Fbr, we prove that for any normal subgroup N of Fbr, either N is contained in the kernel of Fbr \onto F, or else N contains [Fbr,Fbr]. We also compute the Bieri-Neumann-Strebel invariant Sigma^1(Fbr), which is a useful tool for understanding normal subgroups containing the commutator subgroup.
In this paper, we give a characterization of the normal subgroups of index $2^{s}(2n+1),\ s\in\{1,2\},\ n\in \mathbb{N}$ and of the subgroups of index three of the group representation of the Cayley tree.
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.
The maximal normal subgroup growth type of a finitely generated group is $n^{\log n}$. Very little is known about groups with this type of growth. In particular, the following is a long standing problem: Let $Γ$ be a group and $Δ$ a subgroup of finite index. Suppose $Δ$ has normal subgroup growth of type $n^{\log n}$, does $Γ$ has normal subgroup growth of type $n^{\log n}$? We give a positive answer in some cases, generalizing a result of Müller and the second author and a result of Gerdau. For instance, suppose $G$ is a profinite group and $H$ an open subgroup of $G$. We show that if $H$ is …
We partly generalize the estimate for the rank of intersection of subgroups in free products of groups, proved earlier by S.V.Ivanov and W.Dicks, to the case of free amalgamated products of groups with normal finite amalgamated subgroup. We also prove that the obtained estimate is sharp and cannot be further improved when the amalgamated product contains an involution.
A finite group $P$ is said to be \emph{primary} if $|P|=p^{a}$ for some prime $p$. We say a primary subgroup $P$ of a finite group $G$ satisfies the \emph{Frobenius normalizer condition} in $G$ if $N_{G}(P)/C_{G}(P)$ is a $p$-group provided $P$ is $p$-group. In this paper, we determine the structure of a finite group $G$ in which every non-subnormal primary subgroup satisfies the Frobenius normalized condition. In particular, we prove that if every non-normal primary subgroup of $G$ satisfies the Frobenius condition, then $G/F(G)$ is cyclic and every maximal non-normal nilpotent subgroup $U$ o…
On Normal Subgroups of Coxeter Groups Generated by Standard Parabolic Subgroups
本文讨论了Coxeter群范畴中非标准子群的一种构造。给出了这类子群的增长级数的两个公式。作为应用，我们构造了一个旗形单凸多胞体，其f-多项式具有非实根.
On the inheriting of the property $C_π$ by some normal subgroups
本文证明了指数为$π'$-数的正规子群继承Hall性质$C_π$。
In this paper we present some algebraic properties of subgroupoids and normal subgroupoids. We define the normalizer of a wide subgroupoid $\mathcal{H}$ and show that, as in the case of groups, the normalizer is the greatest wide subgroupoid of the groupoid $\mathcal{G}$ in which $\mathcal{H}$ is normal. Furthermore, we give the definition of center and commutator and prove that both are normal subgroupoids, the first one of the union of all the isotropy groups of $\mathcal{G}$ and the second one of $\mathcal{G}$. Finally, we introduce the concept of inner isomorphism of $\mathcal{G}$ and show…
Finite groups with some particular maximal invariant subgroups being nilpotent or all non-nilpotent maximal invariant subgroups being normal
本文分类了在互素自同构群作用下、含某不变 Sylow 子群正规化子的极大不变子群均为幂零的有限群，并证明该条件等价于所有非幂零极大不变子群均正规。
Automorphisms fixing every normal subgroup of a nilpotent-by-abelian group
除其他事项外，我们证明了自同构固定的每一个正规子群的幂零由阿贝尔群是幂零由亚阿贝尔。特别地，固定亚阿贝尔群的每个正规子群的自同构群是导长至多为3的可解群。一个例子表明，这个界限不能被改善。
Three amalgams with remarkable normal subgroup structures
本文构造了三个群$Λ_1$，$Λ_2$，$Λ_3$，它们都可以分解为合并积$F_9 \ast_{F_{81}} F_{9}$，并且只有很少的有限或无限指标的正规子群.具体地说，$Λ_1$是单群，$Λ_2$不是单群但没有无穷指数的非平凡正规子群，$Λ_3$不是单群但没有有限指数的真子群.
In this paper we give full description of normal subgroups of index eight and ten for the group.
It is known that any locally graded group with finitely many derived subgroups of non-normal subgroups is finite-by-abelian. This result is generalized here, by proving that in a locally graded group $G$ the subgroup $γ_{k}(G)$ is finite if the set $\{γ_{k}(H)\;|\;H\ntriangleleft G\}$ is finite. Moreover, locally graded groups with finitely many $k$th terms of lower central series of infinite non-normal subgroups are also completely described.
On open normal subgroups of parahorics
设$F$是具有离散赋值的局部完备域，$G$是$F$上的拟分裂群，它在$F$的某个非分歧扩张上分裂.设$P$是$G$的$F$ -点群$G(F)$的一个对位子群，$P$的开正规亲幂零子群可以用Prasad和Raghanathan的标准正规滤子群来分类.更精确地说，如果$G$是拟单的，并且满足一些附加条件，则$H$是$G$的某个极大环面的模子群，这些滤子群之一或其中之一与$P\cap M$的标准正规滤子群的乘积，其中$M$是$G$的真Levi子群.
Let $G$ be a finite solvable group. We show that $G$ does not have a normal nonabelian Sylow $p$-subgroup when its prime character degree graph $Δ(G)$ satisfies a technical hypothesis.
We show that the product of two partial normal subgroups of a locality (in the sense of Chermak) is again a partial normal subgroup. This generalizes a theorem of Chermak and fits into the context of building a local theory of localities.
A group G is a cn-group if for each subgroup H of G there exists a normal subgroup N of G such that the index of both H and N in HN is finite. The class of cn-groups contains properly the classes of core- finite groups and that of groups in which each subgroup has finite index in a normal subgroup. In the present paper it is shown that a cn-group whose periodic images are locally finite is finite-by-abelian-by-finite. Such groups are then described into some details by considering automorphisms of abelian groups. Finally, it is shown that if G is a locally graded group with the property that t…
A locally normal subgroup in a topological group is a subgroup whose normaliser is open. In this paper, we provide a detailed description of the large-scale structure of closed locally normal subgroups of complete Kac-Moody groups over finite fields. Combining that description with the main result from arXiv:2111.07066, we show that under mild assumptions, if the Kac-Moody group is one-ended (a property that is easily determined from the generalised Cartan matrix), then it is locally indecomposable, which means that no open subgroup decomposes as a nontrivial direct product.
A note on invariant transversals for normal subgroups
本文研究了有限群中阿贝尔正规子群的不变横截面存在性，并给出了相关猜想的反例。
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.
Finite solvable groups with a nilpotent normal complement subgroup
本文研究有限可解群中非正规无核子群的结构，证明在特定正规化子条件下，该子群存在幂零正规补，且相关构造形成Frobenius群。
In [A. Stolz and A. Thom, On the lattice of normal subgroups in ultraproducts of compact simple groups, PLMS 108(1), 2014] it was stated that the lattice of normal subgroups of an ultraproduct of finite simple groups is always linearly ordered. This is false in this form in most cases for classical groups of Lie type. We correct the statement and point out a version of 'relative' bounded generation results for classical quasisimple groups and its implications on the structure of the lattice of normal subgroups of an ultraproduct of quasisimple groups.
In this paper, we describe the automorphism group of semidirect product of two groups that fixes the non-normal subgroup of it. We have computed these automorphisms for the non-abelian metacyclic $p$-group and non-abelian $p$-groups $(p\ge 5)$ of order $p^{4}$, where $p$ is a prime.
In this article we introduce and study a class of finite groups for which the orders of normal subgroups satisfy a certain inequality. It is closely connected to some well-known arithmetic classes of natural numbers.
It is well known that the normaliser of a parabolic subgroup of a finite Coxeter group is the semidirect product of the parabolic subgroup by the stabiliser of a set of simple roots. We show that a similar result holds for all finite unitary reflection groups: namely, the normaliser of a parabolic subgroup is a semidirect product and in many, but not all, cases the complement can be obtained as the stabiliser of a set of roots. In addition we record when the complement acts as a reflection group on the space of fixed points of the parabolic subgroup.
Some Properties of normal subgroups determined from character tables
本文分析了可从G特征标表确定的正规子群结构性质，扩展了Thompson定理和Taketa定理，并引入了幂零类受不可约特征标度数量界定的新类nMI群。
Narrow normal subgroups of Coxeter groups and of automorphism groups of Coxeter groups
本文研究Coxeter群及其自同构群中的正规子群，刻画了其中有限子群与不含非阿贝尔自由群的窄子群的结构。
We show that any group $G$ is contained in some sharply 2-transitive group $\mathcal{G}$ without a non-trivial abelian normal subgroup. This answers a long-standing open question. The involutions in the groups $\mathcal{G}$ that we construct have no fixed points.
The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of $GL(n, K)$ ($K$ - a field, $n \geq 3$) which is not contained in the center, contains $SL(n, K)$. A. Rosenberg gave description of normal subgroups of $GL(V)$, where $V$ is a vector space of any infinite cardinality dimension over a division ring. However, when he considers subgroups of the direct product of the center and the group of linear transformations $g$ such that $g-id_V$ has finite dimensional range the proof is not complete. We fill this gap for countably dimensional $V$ giving description of …
Any virtually free group $H$ containing no non-trivial finite normal subgroup (e.g., the infinite dihedral group) is a retract of any finitely generated group containing $H$ as a verbally closed subgroup.
We consider the structure of a finite groups having a normal series whose factors have bicyclic Sylow subgroups. In particular, we investigated groups of odd order and $A_4$-free groups with this property. Exact estimations of the derived length and nilpotent length of such groups are obtained.
We improve a bound of Borcherds on the virtual cohomological dimension of the non-reflection part of the normalizer of a parabolic subgroup of a Coxeter group. Our bound is in terms of the types of the components of the corresponding Coxeter subdiagram rather than the number of nodes. A consequence is an extension of Brink's result that the non-reflection part of a reflection centralizer is free. Namely, the non-reflection part of the normalizer of parabolic subgroup of type D5 or Aodd is either free or has a free subgroup of index 2.
Normalizers of Sylow subgroups in finite reflection groups
本文证明了有限反射群中西罗子群正规化子的半直积分解，并利用抛物子群与考克斯特图自同构给出实型情形的具体描述。
We establish a general normal subgroup theorem for commensurators of lattices in locally compact groups. While the statement is completely elementary, its proof, which rests on the original strategy of Margulis in the case of higher rank lattices, relies heavily on analytic tools pertaining to amenability and Kazhdan's property (T). It is a counterpart to the normal subgroup theorem for irreducible lattices of Bader and the second named author, and may also be used to sharpen that result when one of the ambient factors is totally disconnected.
A formula for the normal subgroup growth of Baumslag-Solitar groups
当p和q互质时，给出了Baumslag-Solitar群BS（p，q）中每个有限指标的正规子群个数的精确公式.与所有有限指数子群的公式不同，这个公式区分了不同的Baumslag-Solitar群，并且不是乘法的。这使我们能够给出一个例子，一个非线性生成的投射有限群，它不是虚幂零的，但它的zeta函数有一个欧拉积。
New characterizations of a normal subgroup to be hypercyclically embedded
本文研究群的正规子群超循环嵌入条件，基于主因子循环性给出若干新刻画，推广并统一了相关近期结果。
We study the class of groups having the property that every non-nilpotent subgroup is equal to its normalizer. These groups are either soluble or perfect. We completely describe the structure of soluble groups and finite perfect groups with the above property. Furthermore, we give some structural information in the infinite perfect case.
In this note we show that if $p$ is an odd prime and $G$ is a powerful $p$-group with $N\leq G^{p}$ and $N$ normal in $G$, then $N$ is powerfully nilpotent. An analogous result is proved for $p=2$ when $N\leq G^{4}$.
Normal subgroup growth in free class-2-nilpotent groups
设$F_{2,d}$表示$d$生成元上的自由2类幂零群.我们计算了正规zeta函数$ζ^\triangleleft_{F_{2,d}}(s)$，证明了它们满足局部函数方程，并确定了它们的收敛横坐标和极点阶。
We study groups having the property that every non-abelian subgroup is equal to its normalizer. This class of groups is closely related to an open problem posed by Berkovich. We give a full classification of finite groups having the above property. We also describe all infinite soluble groups in this class.
Let G be a group and H be a subgroup of G which is either finite or of finite index in G. In this note, we give some characterizations for normality of H in G. As a consequence we get a very short and elementary proof of the Main Theorem of [5], which avoids the use of the classification of finite simple groups
We investigate the class $\mathcal{MN}$ of groups with the property that all maximal subgroups are normal. The class $\mathcal{MN}$ appeared in the framework of the study of potential counter-examples to the Andrews-Curtis conjecture. In this note we give various structural properties of groups in $\mathcal{MN}$ and present examples of groups in $\mathcal{MN}$ and not in $\mathcal{MN}$.
Normal Subgroup Theorem for groups acting on $\tilde A_2$-buildings
本文证明了作用于$\ ilde A_2$型建筑且具有有限稳定子和有限基本域的群，其任何非平凡正规子群都是有限指数的。
We announce various results concerning the structure of compactly generated simple locally compact groups. We introduce a local invariant, called the structure lattice, which consists of commensurability classes of compact subgroups with open normaliser, and show that its properties reflect the global structure of the ambient group.
Normal subgroups of odd-order monomial $p^a q^b$ groups
有限群$G$称为单群，如果$G$的每个不可约特征标都是由$G$的某个子群的线性特征标导出的.关于单项群的一个主要问题是单项群$G$的正规子群$N$本身是否是单项的。在$G$是偶数阶群的情况下，已经证明（戴德，范德瓦尔），$N$不需要是单项式的。本文证明了，若$G$是$p^aq^b$阶单项式群，其中$p$和$q$是不同的奇素数，则$G$的任何正规子群$N$也是单项式群.
In this article, we prove that if all non-trivial cyclic subgroups of a group $G$ are self normalizing and $G$ satisfies the implication $$ \ o(x)\neq o(y)\Rightarrow o(xy)\neq o(x), o(y), $$ for all non-trivial elements $x$ and $y$, then $G$ is a $p$-group or simple.
Suppose $\mathcal{E}$ is a normal subsystem of a saturated fusion system $\mathcal{F}$ over $S$. If $X\leq S$ is fully $\mathcal{F}$-normalized, then Aschbacher defined a normal subsystem $N_{\mathcal{E}}(X)$ of $N_{\mathcal{F}}(X)$. In this short note we revisit and generalize this result using the theory of localities. Our more general approach leads in particular to a normal subsystem $C_{\mathcal{E}}(X)$ of $C_{\mathcal{F}}(X)$ for every $X\leq S$ which is fully $\mathcal{F}$-centralized.
Let G be a p-adic Lie group and Ad be the adjoint representation of G on its Lie algebra. It was claimed in the literature that the kernel K of Ad always has an abelian open normal subgroup. We show by means of a counterexample that this assertion is false; it can even happen that K=G but G has no abelian subnormal subgroup except for the trivial group. The arguments are based on auxiliary results on subgroups of free products with central amalgamation.
Glauberman's $Z^*$-theorem and analogous statements for odd primes show that, for any prime $p$ and any finite group $G$ with Sylow $p$-subgroup $S$, the centre of $G/O_{p^\prime}(G)$ is determined by the fusion system $\mathcal{F}_S(G)$. Building on these results we show a statement that seems a priori more general: For any normal subgroup $H$ of $G$ with $O_{p^\prime}(H)=1$, the centralizer $C_S(H)$ is expressed in terms of the fusion system $\mathcal{F}_S(H)$ and its normal subsystem induced by $H$.
Characterizing normal Hall subgroups by character degrees, II
本文证明了有限单经典群中真非平凡 Hall 子群的特征标次数性质，从而结合已有约化，给出了任意有限群中 Hall 子群正规性的充要条件：其诱导表示的不可约成分次数均为 $\pi'$ 数。
Lifts of partial characters with respect to a chain of normal subgroups
本文考虑$π$ -部分特征标的提升，其性质是它们对某些正规子群的限制的不可约分支也是提升。我们将证明这样的提升必须从我们称之为归纳对的东西中导出，并且从归纳对导出的每个特征标都是这样的提升。有了这个条件，我们将得到这样的电梯数量的下界。
Groups with a conjugacy class that is the difference of two normal subgroups
研究有限群中存在两个正规子群差作为共轭类的结构，通过特征理论刻画并确定此类群的性质，进一步推广了Gagola的结果。
Quotients of permutation groups by nonabelian minimal normal subgroups
本文证明了 Sym(n) 的子群关于非交换极小正规子群的商可嵌入更小次数的置换群，并在传递情况下给出了次数的上界 m \le 2n/5。
On the normal subgroups of a split extension
本文通过N和Q的正规子群描述半直积G=N⋊Q的正规子群，其中N和Q是互素的有限群，并给出循环Sylow子群的有限群的正规子群完全列表。
Let $G$ be a finite group and $p$ a fixed prime divisor of $|G|$. Combining the nilpotence, the normality and the order of groups together, we prove that if every maximal subgroup of $G$ is nilpotent or normal or has $p'$-order, then (1) $G$ is solvable; (2) $G$ has a Sylow tower; (3) There exists at most one prime divisor $q$ of $|G|$ such that $G$ is neither $q$-nilpotent nor $q$-closed, where $q\neq p$.
Normal subgroups whose conjugacy class graph has diameter three
本文研究了有限群 $G$ 的正规子群 $N$，确定了当包含在 $N$ 中的 $G$-共轭类相关联的图的直径达到最大值三时 $N$ 的结构。
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