Non-isomorphic Subgroups Research Articles

The Abelian Groups of Large Order: Perspective from (Fuzzy) Subgroups of Finite p-Groups

In the recent past, results have shown that Nilpotent groups such as p-groups, have normal series of finite length. Any finite p-group has many normal subgroups and consequently, the phenomenon of large number of non-isomorphic subgroups of a given order. This makes it an ideal object for combinatorial and cohomological investigations. Cartesian product (otherwise known as the product set) plays vital roles in the course of synthesizing the abstract groups. Previous studies have determined the number of distinct fuzzy subgroups of various finite p-groups including those of square-free order. However, not much work has been done on the fuzzy subgroup classification for the nilpotent groups formed from the Cartesian products of p-groups through their computations. Here, part of our intention is therefore trying to make some designs so as to classify the nilpotent groups formed from the Cartesian products of p-groups through their computations. The Cartesian products of p-groups were taken to obtain nilpotent groups. Results up to two dimensions are now obtainable. In this paper, the fuzzy subgroups of the nilpotent product of two abelian subgroups of orders 2^<i>n ^</i>and 128. The integers n ≥ 7 have been successfully considered and the derivation for the explicit formulae for its number distinct fuzzy subgroups were calculated. Some methods were once being used in counting the chains of fuzzy subgroups of an arbitrary finite <i>p</i>-group <I>G. </I>Here, the adoption of the famous Inclusion-Exclusion principle is very necessary and imperative so as to obtain a reasonable, and as much as possible accurate.

The Cayley Property of Some Distant Graphs and Relationship with the Stern\u2013Brocot Tree

One of the graphs associated with any ring R is its distant graph G(R,Delta ) with points of the projective line mathbb {P}(R) over R as vertices. We prove that the distant graph of any commutative, Artinian ring is a Cayley graph. The main result is the fact that G(mathbb Z,Delta ) is a Cayley graph of a non-artinian commutative ring. We indicate two non-isomorphic subgroups of PSL_2(mathbb Z) corresponding to this graph.

Phase-transformation-induced twinning of an iron(III) calix[4]pyrrolidine complex.

The title compound, tetrachlorido-1κCl;2κ(3)Cl-(2,2,7,7,12,12,17,17-octamethyl-21,22,23,24-tetraazapentacyclo[16.2.1.1(3,6).1(8,11).1(13,16)]tetracosane-1κ(4)N,N',N'',N''')-μ2-oxido-diiron(III), [Fe2Cl4O(C28H52N4)], undergoes a slow phase transformation at ca 173 K from monoclinic space group P2(1)/n, denoted form (I), to the maximal non-isomorphic subgroup, triclinic space group P1, denoted form (II), which is accompanied by nonmerohedral twinning [twin fractions of 0.693 (4) and 0.307 (4)]. The transformation was found to be reversible, as on raising the temperature the crystal reverted to monoclinic form (I). In the asymmetric unit of form (I), Z' = 1, while in form (II), Z' = 2, with a very small reduction (ca 1.8%) in the unit-cell volume. The two independent molecules (A and B) in form (II) are related by a pseudo-twofold screw axis along the b axis. The molecular overlay of molecule A on molecule B has an r.m.s. deviation of 0.353 Å, with the largest distance between two equivalent atoms being 1.202 Å. The reaction of calix[4]pyrrolidine, the fully reduced form of meso-octamethylporphyrinogen, with FeCl3 gave a red-brown solid that was recrystallized from ethanol in air, resulting in the formation of the title compound. In both forms, (I) and (II), the Fe(III) atoms are coordinated to the macrocyclic ligand and have distorted octahedral FeN4OCl coordination spheres. These Fe(III) atoms lie out of the mean plane of the four N atoms, displaced towards the O atom of the [OFeCl3] unit by 0.2265 (5) Å in form (I), and by 0.2210 (14) and 0.2089 (14) Å, respectively, in the two independent molecules (A and B) of form (II). The geometry of the [OFeCl3] units are similar, with each Fe(III) atom having a tetrahedral coordination sphere. The NH H atoms are directed below the planes of the macrocycles and are hydrogen bonded to the coordinated Cl(-) ions. There are also intramolecular C-H···Cl hydrogen bonds present in both (I) and (II). In form (I), there are no significant intermolecular interactions present. In form (II), the individual molecules are arranged in alternate layers parallel to the ac plane. The B molecules are linked by a C-H···Cl hydrogen bond, forming chains along [100].

Gold Network Structures in Rhombohedral and Monoclinic Sr2Au6(Au,T)3 (T = Zn, Ga). A Transition via Relaxation

Quantitative syntheses, structure determinations and interpretations, and band calculations are reported for the nonstoichiometric rhombohedral (R3c) and monoclinic (C2/c) Sr2Au6(Au(3-x)T(x)) (T = Zn, Ga) compounds. Several different compositions of the two Sr phases were similarly refined from single crystal X-ray diffraction data as R3c: a ≈ 8.43 Å, c ≈ 21.85 Å, Z = 6 and C2/c: a ≈ 14.70 Å, b ≈ 8.47 Å, c ≈ 8.70 Å, β ≈ 123.2°, Z = 4. The R3c Zn phase is stable in the composition region x ∼ 2.5-2.9 whereas its C2/c neighbor is the major product at x ∼ 2.2-2.3. Gallium versions of both were also identified. Both R3c and C2/c structural types contain hexagonal-diamond-like gold superlattices stuffed with strings of interstitial Sr and disordered triangular (Au,T)3 units. The latter space group is a maximal, nonisomorphic subgroup of the former, and the decrease in interstitial radius from Ba to Sr (∼0.08 Å experimentally) evidently drives the symmetry reduction, relaxation, and small distortions, principally around the Sr sites. Au-Au bonding among the Au hexagons in the host lattices and with gold components in the triangular interstitials is dominant and reflected in their tight packing and short interatomic separations.

On the crystal structures of Ln3MO 7 ( Ln=Nd, Sm, Y and M=Sb, Ta)—Rietveld refinement using X-ray powder diffraction data

We have investigated, using X-ray powder diffraction data, the crystal structures of some fluorite derivatives with the formula Ln 3 MO 7 ( Ln=lanthanide or Y and M=Sb and Ta). In these compounds ordering of Ln and M occurs, leading to a parent structure in Cmmm. Tilting of the MO 6 octahedra causes doubling of one of the cubic axes, leading to a number of non-isomorphic subgroups, e.g. Cmcm, Ccmm and Cccm. We have identified an alternative space group Ccmm instead of C222 1 for those compounds containing a medium sized lanthanide or Y and M being Sb or Ta. Interestingly this is an alternative setting for the space group of the structure obtained when Ln is large ( Cmcm). However, there tilting of the octahedra is around the a-axis of the parent structure, rather than around the b-axis as it is found in the compounds which we are reporting on here. In one compound, Nd 3TaO 7, both tilts occur. The phase transition between the two possible structures is a slow and difficult process above 80 K, allowing both phases to coexist.

Pseudomorphic 2A→ 2M→ 2H phase transitions in lanthanum strontium germanate electrolyte apatites

Apatite-like materials are of considerable interest as potential solid oxide fuel cell electrolytes, although their structural vagaries continue to attract significant discussion. Understanding these features is crucial both to explain the oxide ion conduction process and to optimise it. As the composition of putative P6(3)/m apatites with ideal formula [A(I)(4)][A(II)(6)][(BO(4))(6)][X](2) is varied the [A(I)(4)(BO(4))(6)] framework will flex to better accommodate the [A(II)(6)X(2)] tunnel component through adjustment of the A(I)O(6) metaprism twist angle (varphi). The space group theory prescribes that framework adaptation during phase changes must lead to one of the maximal non-isomorphic subgroups of P6(3)/m (P2(1), P2(1)/m, P1[combining macron]). These adaptations correlate with oxygen ion conduction, and become crucial especially when the tunnels are filled by relatively small ions and/or partially occupied, and if interstitial oxygens are located in the framework. Detecting and completely describing these lower symmetry structures can be challenging, as it is difficult to precisely control apatite stoichiometry and small departures from the hexagonal metric may be near the limits of detection. Using a combination of diffraction and spectroscopic techniques it is shown that lanthanum strontium germanate oxide electrolytes crystallise as triclinic (A), monoclinic (M) and hexagonal (H) bi-layer pseudomorphs with the composition ranges: [La(10-x)Sr(x)][(GeO(4))(5+x/2)(GeO(5))(1-x/2)][O(2)] (0 <or=x<or= 1) apatite-2A[La(10-x)Sr(x)][(GeO(4))(5+x/2)(GeO(5))(1-x/2)][O(2)] (1 <or=x<or= 2) apatite-2M[La(10-x)Sr(x)][(GeO(4))(6)][O(2)][H(delta)] (2 <or=x<or= 2.96) apatite-2M[La(10-x)Sr(x)][(GeO(4))(6)][O(2)][H(delta)] (2.96 <or=x<or= 5.32) apatite-2HFurthermore, at typical fuel cell operating temperatures apatite-2A and apatite-2M will transform to apatite-2H, with the latter showing the highest conduction. The results show that small twist angles and high symmetry enhance oxygen mobility with these properties tailored by adjusting the relative size of the framework to tunnel. This information can hence aid in the design of new materials with improved oxide ion conductivity.

Rietveld structure refinement of perovskite and post-perovskite phases of NaMgF3 (Neighborite) at high pressures

Neighborite (NaMgF 3 ) with the perovskite structure, transforms to a post-perovskite (ppv) phase between 27 and 30 GPa. The ppv phase is observed to the highest pressures achieved (56 GPa) at room temperature and transforms to an as yet unknown phase upon heating. Rietveld structure refinement using monochromatic synchrotron X-ray diffraction data provide models for the perovskite and post-perovskite structures at high pressure. The refined models at 27(1) GPa indicate some inter-octahedral F-F distances rival the average intra-octahedral distance, which may cause instability in the perovskite structure and drive the transformation to the post-perovskite phase. The ratio of A-site to B-site volume ( V A / V B ) in perovskite structured NaMgF 3 (ABX 3 ), spans from 5 in the zero-pressure high-temperature cubic perovskite phase to 4 in this high-pressure perovskite phase at 27(1) GPa, matching the V A / V B value in post-perovskite NaMgF 3 . Using Rietveld refinement on post-perovskite structure models, we observe discrepancies in pattern fitting, which may be described in terms of development of sample texture in the diamond-anvil cell, recrystallization, or a change of space group to Cmc 2 1 , a non-isomorphic subgroup of Cmcm —the space group describing the structure of CaIrO 3 .

Special Families of Sets and Baer-Specker Groups#

ABSTRACT We prove that the Baer-Specker group Π = contains a pure subgroup isomorphic to the direct sum of 2 0 copies of itself. We produce 2 ℵ 0 nonisomorphic subgroups of Π, each isomorphic to its dual group. Finally, we show that the isomorphism type of a generalized product of ℤ's, the set of functions I → ℤ with support of size at most α, uniquely determines both the cardinality of I and α (as long as there are no measurable cardinals ≤α). All three of these results are obtained using set-theoretic existence theorems, namely, the existence of large independent families, large almost disjoint families, and Δ-systems.

Phase transitions in compounds with the Ca3Ga2Ge4O14 structure

Possible structural changes described by the group-subgroup relationships in the Ca3Ga2Ge4O14-type structure (sp. gr. P321) are considered. The most probable phase transitions seem to be those accompanied by lowering of the symmetry to the maximal non-isomorphic subgroups P3 and C2. It is shown that only destructive phase transitions accompanied by symmetry rise up to the minimal non-isomorphic supergroups for the given structure type can take place. The change of the trigonal symmetry to monoclinic is revealed in La3SbZn3Ge2O14, whose crystal structure is refined as a derivative structure of the Ca3Ga2Ge4O14 structure type within the sp. gr. A2 (C2). At ∼250°C, La3SbZn3Ge2O14 undergoes a reversible phase transition accompanied by symmetry rise, A2 ⇄ P321. Similar phase transitions, P321 ⇄ A2, are also observed in La3Nb0.5Ga5.5O14 and La3Ta0.5Ga5.5O14 under the hydrostatic pressures 12.4(3) and 11.7(3) GPa, respectively. The mechanisms of compression and phase transition are based on the anisotropic compressibility of a layer structure. With the attainment of the critical stress level in the structure, the elevated compressibility in the (ab) plane gives rise to a phase transition accompanied by the loss of the threefold axis. Attempts to reveal low-temperature phase transitions in a number of representatives of the langasite family have failed.

Phasing protein structures using the group-subgroup relation.

Diffraction data from two non-isomorphous crystals (forms 1 and 2) of an artificial protein with a four-helix bundle motif, di-Co(II)-DF1-L13A, have been collected using synchrotron radiation. The phase of form 1 has been assigned using the group and minimal non-isomorphic supergroup relation between the space group of the previously determined di-Mn(II)-DF1-L13G structure and the space group of this form. This unconventional method of solving the phase problem has also been tested with form 2 using a reverse relation. The structure of the latter form has been solved using the group and maximal non-isomorphic subgroup relation with the space group of form 2 of the analogous dimanganese protein. This application has shown that this phasing method can be used for solving the protein structures of polymorphic crystals as an alternative to the molecular-replacement method.

The subgroup-subfactor

In this paper, we compute the standard invariant of the 'subgroup-subfactor' P× α|H H⊂P× α G, where α denotes an outer action of a finite group G on a II 1 factor P, and P× α|H H denotes the obvi- ous crossed-product obtained by restricting the action to H. We then use this description to exhibit a pair of non-isomorphic subgroups H i , i=1, 2, of the symmetric group S 4 such that the subfactors R× α|H i H i ⊂P× α G, i=1, 2 are conjugate, thereby disproving a conjecture of Thomsen-see [9]-that 'the subgroup-subfactor re- members the subgroup' (provided the subgroup contains no non-trivial normal subgroup of the ambient group).

CoReO 4, a new rutile-type derivative with ordering of two cations

CoReO 4 crystallizes in space group Cmmm with a = 6.5156(7)Å, b = 6.7418(8)Å, c = 2.8923(3)Å. It is most likely to be formulated as Co +3Re +5O 4. The bonding topology of the connections of the coordination octahedra around the Co and the Re atoms is identical to the arrangement of the coordination octahedra in the rutile type. Octahedral edge-sharing rutile-type chains of compositions CoO 4 and ReO 4 are present which are connected via corners, resulting in an overall chemical formula of CoReO 4. The space group type of CoReO 4, Cmmm, is a maximal nonisomorphic subgroup of order two of the space group type of rutile, P4 2/mnm; it is translationengleich with its supergroup. The space group type of MgUO 4, Imam, is in turn a maximal nonisomorphic subgroup of order two of space group Cmmm; it is klassengleich with its supergroup. Thus the structure type of CoReO 4 can be seen as an intermediate step in the reduction of the symmetry of the rutile type proper to the distorted rutile-type variant represented by MgUO 4. The crystal structure of CoReO 4 represents a new type derived from the rutile type by ordering of two cations. It is the fourth structure type of this kind.

The Free Metabelian Group of Rank Two Contains Continuously Many Nonisomorphic Subgroups

The Free Metabelian Group of Rank Two Contains Continuously Many Nonisomorphic Subgroups

The free metabelian group of rank two contains continuously many nonisomorphic subgroups

It is proved that the free metabelian group of rank two contains continuously many nonisomorphic subgroups.

Classifying torsion-free subgroups of the Picard group

Torsion-free subgroups of finite index in the Picard group are the fundamental groups of hyperbolic 3 3 -manifolds. The Picard group is a polygonal product of finite groups. Recent work by Karrass, Pietrowski and Solitar on the subgroups of a polygonal product make it feasible to calculate all the torsion-free subgroups of any finite index. This computation is carried out here for index 12 and 24, where there are, respectively, 2 and 17 nonisomorphic subgroups. The manifolds are identified by using surgery.

3357. Isomorphic factor groups from non-isomorphic subgroups

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