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Search: All articles in the CMB digital archive with keyword nilpotent group

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1. CMB 2016 (vol 60 pp. 12)

Akbari, Saieed; Miraftab, Babak; Nikandish, Reza
Co-maximal Graphs of Subgroups of Groups
Let $H$ be a group. The co-maximal graph of subgroups of $H$, denoted by $\Gamma(H)$, is a graph whose vertices are non-trivial and proper subgroups of $H$ and two distinct vertices $L$ and $K$ are adjacent in $\Gamma(H)$ if and only if $H=LK$. In this paper, we study the connectivity, diameter, clique number and vertex chromatic number of $\Gamma(H)$. For instance, we show that if $\Gamma(H)$ has no isolated vertex, then $\Gamma(H)$ is connected with diameter at most $3$. Also, we characterize all finite groups whose co-maximal graphs are connected. Among other results, we show that if $H$ is a finitely generated solvable group and $\Gamma(H)$ is connected and moreover the degree of a maximal subgroup is finite, then $H$ is finite. Furthermore, we show that the degree of each vertex in the co-maximal graph of a general linear group over an algebraically closed field is zero or infinite.

Keywords:co-maximal graphs of subgroups of groups, diameter, nilpotent group, solvable group
Categories:05C25, 05E15, 20D10, 20D15

2. CMB 2016 (vol 60 pp. 77)

Christ, Michael; Rieffel, Marc A.
Nilpotent Group C*-algebras as Compact Quantum Metric Spaces
Let $\mathbb{L}$ be a length function on a group $G$, and let $M_\mathbb{L}$ denote the operator of pointwise multiplication by $\mathbb{L}$ on $\lt(G)$. Following Connes, $M_\mathbb{L}$ can be used as a ``Dirac'' operator for the reduced group C*-algebra $C_r^*(G)$. It defines a Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of $C_r^*(G)$. We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-$*$ topology (a key property for the definition of a ``compact quantum metric space''). In particular, this holds for all word-length functions on finitely generated nilpotent-by-finite groups.

Keywords:group C*-algebra, Dirac operator, quantum metric space, discrete nilpotent group, polynomial growth
Categories:46L87, 20F65, 22D15, 53C23, 58B34

3. CMB 2014 (vol 57 pp. 884)

Xu, Yong; Zhang, Xinjian
$m$-embedded Subgroups and $p$-nilpotency of Finite Groups
Let $A$ be a subgroup of a finite group $G$ and $\Sigma : G_0\leq G_1\leq\cdots \leq G_n$ some subgroup series of $G$. Suppose that for each pair $(K,H)$ such that $K$ is a maximal subgroup of $H$ and $G_{i-1}\leq K \lt H\leq G_i$, for some $i$, either $A\cap H = A\cap K$ or $AH = AK$. Then $A$ is said to be $\Sigma$-embedded in $G$; $A$ is said to be $m$-embedded in $G$ if $G$ has a subnormal subgroup $T$ and a $\{1\leq G\}$-embedded subgroup $C$ in $G$ such that $G = AT$ and $T\cap A\leq C\leq A$. In this article, some sufficient conditions for a finite group $G$ to be $p$-nilpotent are given whenever all subgroups with order $p^{k}$ of a Sylow $p$-subgroup of $G$ are $m$-embedded for a given positive integer $k$.

Keywords:finite group, $p$-nilpotent group, $m$-embedded subgroup
Categories:20D10, 20D15

4. CMB 2013 (vol 57 pp. 125)

Mlaiki, Nabil M.
Camina Triples
In this paper, we study Camina triples. Camina triples are a generalization of Camina pairs. Camina pairs were first introduced in 1978 by A .R. Camina. Camina's work was inspired by the study of Frobenius groups. We show that if $(G,N,M)$ is a Camina triple, then either $G/N$ is a $p$-group, or $M$ is abelian, or $M$ has a non-trivial nilpotent or Frobenius quotient.

Keywords:Camina triples, Camina pairs, nilpotent groups, vanishing off subgroup, irreducible characters, solvable groups

5. CMB 2001 (vol 44 pp. 266)

Cencelj, M.; Dranishnikov, A. N.
Extension of Maps to Nilpotent Spaces
We show that every compactum has cohomological dimension $1$ with respect to a finitely generated nilpotent group $G$ whenever it has cohomological dimension $1$ with respect to the abelianization of $G$. This is applied to the extension theory to obtain a cohomological dimension theory condition for a finite-dimensional compactum $X$ for extendability of every map from a closed subset of $X$ into a nilpotent $\CW$-complex $M$ with finitely generated homotopy groups over all of $X$.

Keywords:cohomological dimension, extension of maps, nilpotent group, nilpotent space
Categories:55M10, 55S36, 54C20, 54F45

6. CMB 1999 (vol 42 pp. 335)

Kim, Goansu; Tang, C. Y.
Cyclic Subgroup Separability of HNN-Extensions with Cyclic Associated Subgroups
We derive a necessary and sufficient condition for HNN-extensions of cyclic subgroup separable groups with cyclic associated subgroups to be cyclic subgroup separable. Applying this, we explicitly characterize the residual finiteness and the cyclic subgroup separability of HNN-extensions of abelian groups with cyclic associated subgroups. We also consider these residual properties of HNN-extensions of nilpotent groups with cyclic associated subgroups.

Keywords:HNN-extension, nilpotent groups, cyclic subgroup separable $(\pi_c)$, residually finite
Categories:20E26, 20E06, 20F10

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