R. Baer证明了：如果群G的上中心序列的成员γ_{n}（G）构成的群G/γ_{n}（G）是有限的（这里n是正整数），则群G的下中心序列的成员γ_{n+1}（G）也是有限的。特别地，在这种情况下，G的幂零剩余是有限的。这个定理承认最近由M. de Falco，F. de Giovanni，C.穆塞拉和雅P. Sysak：“若群G的模其上超中心Z的因子群G/Z是有限的，则G有有限正规子群L使得G/L是超中心的”。在当前的文章中，我们提供了这个定理的一个新的更简单的非常简短的证明，并详细说明了它。事实上，我们证明，如果|G/Z| =t则|L| \leqt^{k}，其中k=（1/2）（log_{p} t+1），p是t的最小素除子。
英文摘要：
R. Baer has proved that if the factor-group G/ζ_{n}(G) of a group G by the member ζ_{n}(G) of its upper central series is finite (here n is a positive integer) then the member γ_{n+1}(G) of the lower central series of G is also finite. In particular, in this case, the nilpotent residual of G is finite. This theorem admits the following simple generalization that has been published very recently by M. de Falco, F. de Giovanni, C. Musella and Ya. P. Sysak: "If the factor-group G/Z of a group G modulo its upper hypercenter Z is finite then G has a finite normal subgroup L such that G/L is hypercentral". In the current article we offer a new simpler very short proof of this theorem and specify it substantially. In fact, we prove that if |G/Z|=t then |L|\leqt^{k}, where k=(1/2)(log_{p}t+1), and p is the least prime divisor of t.
In the paper we prove (modulo the classification of finite simple groups) an analogue of the famous Baer-Suzuki theorem for the $π$-radical of a finite group, where $π$ is a set of primes
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.
We show that if a group $G$ has a finite normal subgroup $L$ such that $G/L$ is hypercentral, then the index of the hypercenter of $G$ is bounded by a function of the order of $L$. This completes recent results generalizing classical theorems by R. Baer and P. Hall. Then we apply our results to groups of automorphisms of a group $G$ acting in a restricted way on an ascending normal series of $G$.
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 …
We give group analogs of two important theorems of real algebra concerning convex valuations, one of which is the Baer-Krull theorem. We do this by using quasi-orders, which gives a uniform approach to valued and ordered groups. We also recover the classical Baer-Krull theorem from its group analog.
Let $π$ be a proper subset of the set of all primes. Denote by $r$ the smallest prime which does not belong to $π$ and set $m = r$ if $r = 2$ or $3$ and $m = r-1$ if $r \geqslant 5$. We study the following conjecture: a conjugacy class $D$ of a finite group $G$ is contained in the $π$-radical $\mathrm{O}_π(G)$ of $G$ if and only if every $m$ elements of $D$ generate a $π$-subgroup. We confirm this conjecture for each group $G$ whose nonabelian composition factors are isomorphic to sporadic or alternating groups.
Schur's Theorem and its generalisation, Baer's Theorem, are distinguished results in group theory, connecting the upper central quotients with the lower central series. The aim of this paper is to generalise these results in two different directions, using novel methods related with the non-abelian tensor product. In particular, we prove a version of Schur-Baer Theorem for finitely generated groups. Then, we apply these newly obtained results to describe the $k$-nilpotent multiplier, for $k\geq 2$, and other invariants of groups.
Let $π$ be a set of primes such that $|π|\geqslant 2$ and $π$ differs from the set of all primes. Denote by $r$ the smallest prime which does not belong to $π$ and set $m=r$ if $r=2,3$ and $m=r-1$ if $r\geqslant 5$. We study the following conjecture: a conjugacy class $D$ of a finite group $G$ is contained in $Oπ(G)$ if and only if every $m$ elements of $D$ generate a $π$-subgroup. We confirm this conjecture for each group $G$ whose nonabelian composition factors are isomorphic to alternating, linear and unitary simple groups.
Subgroup Theorems for the Baer-invariant of Groups
M. R. Jones和J.Wiegold在[3]中证明了：如果$G$是一个具有有限指标$n$的子群$H$的有限群，则$G$的Schur乘子的$n$次幂$M(G)^n$同构于$M(H)$的一个子群.本文证明了一个类似的结果，中心由中心由$w$群簇，其中$w$是任何外部交换子字。然后利用M.R.R. Moghabu [6]的一个结果，当$c+1$为任意素数或4时，我们将能够推出关于至多为$c$ $(c\geq 1)$类的幂零群簇的Schur型结果（见[4，9]）。
It is shown that the Baer-Kaplansky theorem can be extended to all abelian groups provided that the rings of endomorphisms of groups are replaced by trusses of endomorphisms of corresponding heaps. That is, every abelian group is determined up to isomorphism by its endomorphism truss and every isomorphism between two endomorphism trusses associated to some abelian groups $G$ and $H$ is induced by an isomorphism between $G$ and $H$ and an element from $H$. This correspondence is then extended to all modules over a ring by considering heaps of modules. It is proved that the truss of endomorphism…
M. R. R.莫纳什（Monatsh）90（1980）37-43.）表明，贝尔不变commutes与直接限制的有向系统的群体。本文利用Schur公式对Schur-Baer簇$\mathcal{V}$的$\mathcal{V}$ -覆盖群结构的推广，证明了$\mathcal{V}$ -覆叠群的结构在某种意义下与有向系统的正极限交换.它有一个有用的应用，以推广的$\mathcal{V}$ -覆叠群的几个著名的产品的任意一族群的一些已知的结构。
In the paper autonilpotent groups were characterized as groups $G$ such that $\mathrm{Aut}G$ stabilizes some chain of subgroups of $G$. It was shown that a $p$-group is autonilpotent if and only if its group of automorphisms is also a $p$-group. Analogues of Baer's theorem about the hypercenter and Frobenius $p$-nilpotency criterion were obtained for autonilpotent groups.
Let $ σ$ be a partition of the set of all primes and $\mathfrak{F}$ be a hereditary formation. We described all formations $\mathfrak{F}$ for which the $\mathfrak{F}$-hypercenter and the intersection of weak $K$-$\mathfrak{F}$-subnormalizers of all Sylow subgroups coincide in every group. In particular the formation of all $σ$-nilpotent groups has this property. With the help of our results we solve a particular case of L.A.~Shemetkov's problem about the intersection of $\mathfrak{F}$-maximal subgroups and the $\mathfrak{F}$-hypercenter. As corollaries we obtained P. Hall's and R. Baer's class…
Engel-like characterization of radicals in finite dimensional Lie algebras and finite groups
R. Baer将有限群G的幂零根描述为所有恩格尔元素的集合，即G中的元素y使得对于G中的任何x，第n个交换子[x，y，.，y]等于1，因为n足够大。我们得到了定义在特征为零的域上的有限维李代数的可解根的一个特征。我们提出了一个将有限群的可解根描述为Engel-like元的集合的猜想，并将这个猜想推广到有限单群的情形。
A generalization of von Staudt's theorem that every permutation of the projective line that preserves harmonic quadruples is a projective semilinear map is given. It is then concluded that any proper supergroup of permutations of the projective semilinear group over an algebraically closed field of transcendence degree at least 1 is 4-transitive.
