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Register a new userOn a generalisation of finite $T$-groups
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Let $sigma ={sigma_i |iin I}$ is some partition of all primes $mathbb{P}$ and $G$ a finite group. A subgroup $H$ of $G$ is said to be $sigma$-subnormal in $G$ if there exists a subgroup chain $H=H_0leq H_1leq cdots leq H_n=G$ such that either $H_{i-1}$ is normal in $H_i$ or $H_i/(H_{i-1})_{H_i}$ is a finite $sigma_j$-group for some $j in I$ for $i = 1, ldots, n$. We call a finite group $G$ a $T_{sigma}$-group if every $sigma$-subnormal subgroup is normal in $G$. In this paper, we analyse the structure of the $T_{sigma}$-groups and give some characterisations of the $T_{sigma}$-groups.
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Let $sigma ={sigma_{i} | iin I}$ be some partition of the set of all primes $Bbb{P}$ and let $G$ be a finite group. Then $G$ is said to be $sigma $-full if $G$ has a Hall $sigma _{i}$-subgroup for all $i$. A subgroup $A$ of $G$ is said to be $sigma$-permutable in $G$ provided $G$ is $sigma $-full and $A$ permutes with all Hall $sigma _{i}$-subgroups $H$ of $G$ (that is, $AH=HA$) for all $i$. We obtain a characterization of finite groups $G$ in which $sigma$-permutability is a transitive relation in $G$, that is, if $K$ is a ${sigma}$-permutable subgroup of $H$ and $H$ is a ${sigma}$-permutable subgroup of $G$, then $K$ is a ${sigma}$-permutable subgroup of $G$.
A theorem of Dolfi, Herzog, Kaplan, and Lev cite[Thm.~C]{DHKL} asserts that in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order, and that the inequality is sharp. Inspired by this result and some of the arguments in cite{DHKL}, we establish the following generalisation: if $mathfrak{X}$ is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and $overline{mathfrak{X}}$ is the extension-closure of $mathfrak{X}$, then there exists an (optimal) constant $gamma$ depending only on $mathfrak{X}$ such that, for all non-trivial finite groups $G$ with trivial $mathfrak{X}$-radical, begin{equation} leftlvert G^{overline{mathfrak{X}}}rightrvert ,>, vert Gvert^gamma, end{equation} where $G^{overline{mathfrak{X}}}$ is the ${overline{mathfrak{X}}}$-residual of $G$. When $mathfrak{X} = mathfrak{N}$, the class of finite nilpotent groups, it follows that $overline{mathfrak{X}} = mathfrak{S}$, the class of finite soluble groups, thus we recover the original theorem of Dolfi, Herzog, Kaplan, and Lev. In the last section of our paper, building on J.,G. Thompsons classification of minimal simple groups, we exhibit a family of subgroup-closed Fitting formations $mathfrak{X}$ of full characteristic such that $mathfrak{S} subset overline{mathfrak{X}} subset mathfrak{E}$, thus providing applications of our main result beyond the reach of cite[Thm.~C]{DHKL}.
We construct several series of explicit presentations of infinite hyperbolic groups enjoying Kazhdans property (T). Some of them are significantly shorter than the previously known shortest examples. Moreover, we show that some of those hyperbolic Kazhdan groups possess finite simple quotient groups of arbitrarily large rank; they constitute the first known specimens combining those properties. All the hyperbolic groups we consider are non-positively curved k-fold generalized triangle groups, i.e. groups that possess a simplicial action on a CAT(0) triangle complex, which is sharply transitive on the set of triangles, and such that edge-stabilizers are cyclic of order k.
A finite group $G$ is called a Schur group, if any Schur ring over $G$ is associated in a natural way with a subgroup of $Sym(G)$ that contains all right translations. Recently, the authors have completely identified the cyclic Schur groups. In this paper it is shown that any abelian Schur group belongs to one of several explicitly given families only. In particular, any non-cyclic abelian Schur group of odd order is isomorphic to $Z_3times Z_{3^k}$ or $Z_3times Z_3times Z_p$ where $kge 1$ and $p$ is a prime. In addition, we prove that $Z_2times Z_2times Z_p$ is a Schur group for every prime $p$.
Let $G$ be a simple algebraic group of type $G_2$ over an algebraically closed field of characteristic $2$. We give an example of a finite group $Gamma$ with Sylow $2$-subgroup $Gamma_2$ and an infinite family of pairwise non-conjugate homomorphisms $rhocolon Gammarightarrow G$ whose restrictions to $Gamma_2$ are all conjugate. This answers a question of Burkhard Kulshammer from 1995. We also give an action of $Gamma$ on a connected unipotent group $V$ such that the map of 1-cohomologies ${rm H}^1(Gamma,V)rightarrow {rm H}^1(Gamma_p,V)$ induced by restriction of 1-cocycles has an infinite fibre.
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