Structure of intersection graphs

<p style="text-align: left;" dir="ltr"> Let <em>G</em> be a finite group and let <em>N</em> be a fixed normal subgroup of <em>G</em>. In this paper, a new kind of graph on <em>G</em>, namely the intersection graph is defined and studied. We use <img src="/public/site/images/ikhsan/equation.png" alt="" width="6" height="4" /> to denote this graph, with its vertices are all normal subgroups of <em>G</em> and two distinct vertices are adjacent if their intersection in <em>N</em>. We show some properties of this graph. For instance, the intersection graph is a simple connected with diameter at most two. Furthermore we give the graph structure of <img src="/public/site/images/ikhsan/equation_(1).png" alt="" width="6" height="4" /> for some finite groups such as the symmetric, dihedral, special linear group, quaternion and cyclic groups. </p>

Corneal ectasia in mothers of Down syndrome children

In this study, corneal findings regarding keratoconus (KC) and early KC among mothers with Down syndrome children (MDS) and a group of age-at-delivery-matched mothers with normal children (MNC) were compared. KC was diagnosed based on the presence of a clinical sign and at least one abnormal tomographic or biomechanical criterion. Early KC was defined as having no clinical sign in the presence of at least one abnormal tomographic or biomechanical criterion. The normal subgroups in each group were compared in terms tomographic and biomechanical parameters. In MDS and MNC, the prevalence rates were 6.5% and 1.6% for KC (P = 0.047), and 30.9% and 14.3% for early KC (P = 0.014), respectively. Comparison between the two normal subgroups showed significant differences in mean index of height asymmetry, irregularity index, anterior asphericity, pentacam random forest index, corneal stiffness parameters at first applanation, deformation amplitude ratios, integrated radius-1 mm, highest concavity deflection amplitude, biomechanical corrected IOP, peak distance, and radius (all P < 0.05). This study showed that MDS are more likely to have KC and also to have thinner, steeper and softer corneas compared to MNC. This results support the need for further work for determining the risk of delivering a child with DS.

Cannon–Thurston maps for $${{\,\mathrm{CAT}\,}}(0)$$ groups with isolated flats

Mahan Mitra (Mj) proved Cannon–Thurston maps exist for normal hyperbolic subgroups of a hyperbolic group (Mitra in Topology, 37(3):527–538, 1998). We prove that Cannon–Thurston maps do not exist for infinite normal hyperbolic subgroups of non-hyperbolic $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups with isolated flats with respect to the visual boundaries. We also show Cannon–Thurston maps do not exist for infinite infinite-index normal $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) subgroups with isolated flats in non-hyperbolic $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups with isolated flats. We obtain a structure theorem for the normal subgroups in these settings and show that outer automorphism groups of hyperbolic groups have no purely atoroidal $$\mathbb {Z}^2$$ Z 2 subgroups.

\(S\)-norms on anti \(Q\)-fuzzy subgroups

In this paper, by using \(S\)-norms, we defined anti fuzzy subgroups and anti fuzzy normal subgroups which are new notions and considered their fundamental properties and also made an attempt to study the characterizations of them. Next we investigated image and pre image of them under group homomorphisms. Finally, we introduced the direct sum of them and proved that direct sum of any family of them is also anti fuzzy subgroups and anti fuzzy normal subgroups under \(S\)-norms, respectively.

Normal subgroups of SimpHAtic groups

A group is SimpHAtic if it acts geometrically on a simply connected simplicially hereditarily aspherical (SimpHAtic) complex. We show that finitely presented normal subgroups of the SimpHAtic groups are either: finite, or of finite index, or virtually free. This result applies, in particular, to normal subgroups of systolic groups. We prove similar strong restrictions on group extensions for other classes of asymptotically aspherical groups. The proof relies on studying homotopy types at infinity of groups in question. We also show that non-uniform lattices in SimpHAtic complexes (and in more general complexes) are not finitely presentable and that finitely presented groups acting properly on such complexes act geometrically on SimpHAtic complexes. In Appendix we present the topological two-dimensional quasi-Helly property of systolic complexes.

Normal subgroups in the group of column-finite infinite matrices

The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of GL ⁢ ( n , K ) \mathrm{GL}(n,K) , where 𝐾 is a field and n ≥ 3 n\geq 3 , which is not contained in the center contains SL ⁢ ( n , K ) \mathrm{SL}(n,K) . Rosenberg described the normal subgroups of GL ⁢ ( V ) \mathrm{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 g-\mathrm{id}_{V} has finite-dimensional range, the proof is incomplete. We fill this gap for countably dimensional 𝑉 giving description of the lattice of normal subgroups in the group of infinite column-finite matrices indexed by positive integers over any field.

Normalish Amenable Subgroups of the R. Thompson Groups

Results in [Formula: see text] algebras, of Matte Bon and Le Boudec, and of Haagerup and Olesen, apply to the R. Thompson groups [Formula: see text]. These results together show that [Formula: see text] is non-amenable if and only if [Formula: see text] has a simple reduced [Formula: see text]-algebra. In further investigations into the structure of [Formula: see text]-algebras, Breuillard, Kalantar, Kennedy, and Ozawa introduce the notion of a normalish subgroup of a group [Formula: see text]. They show that if a group [Formula: see text] admits no non-trivial finite normal subgroups and no normalish amenable subgroups then it has a simple reduced [Formula: see text]-algebra. Our chief result concerns the R. Thompson groups [Formula: see text]; we show that there is an elementary amenable group [Formula: see text] [where here, [Formula: see text]] with [Formula: see text] normalish in [Formula: see text]. The proof given uses a natural partial action of the group [Formula: see text] on a regular language determined by a synchronising automaton in order to verify a certain stability condition: once again highlighting the existence of interesting intersections of the theory of [Formula: see text] with various forms of formal language theory.