Ree Group Research Articles

Late Neoarchean monzogranitic–syenogranitic gneisses in the Eastern Hebei–Western Liaoning Province, North China Craton: Petrogenesis and implications for tectonic setting

The medium-grained monzogranitic–syenogranitic gneisses expose as roughly northeast–southwest-trending and cover ∼1500km2 in the southern part of the Eastern Hebei–Western Liaoning Province, North China Craton. These plutonic gneisses have leucocratic features and weakly gneissic to massive structures, and were metamorphosed under greenschist to lower amphibolite facies. LA-ICP-MS zircon U-Pb isotopic dating reveals that the magmatic precursors of these monzogranitic–syenogranitic gneisses were emplaced during 2527–2511Ma. They are characterized by high contents of SiO2 (71.64–78.08%) and K2O (3.55–6.40%), and low contents of CaO (0.25–1.71%), MgO (0.01–0.42%), and Na2O (2.29–4.96%). According to their chondrite-normalized REE patterns, the monzogranitic–syenogranitic gneisses can be subdivided into a high REE group and a low REE group. Studies of petrogenesis reveal that magmatic precursors of the high REE group samples were derived from partial melting of the regionally widespread porphyritic or medium-grained dioritic–quartz monzonitic–granodioritic–monzogranitic gneisses under low pressure conditions (less than ca. 10kbar). The magmatic precursors of the low REE group were formed by fractionated crystallization of the high REE group magma, with titanite, allanite, epidote, apatite, and zircon as the major crystallization fractionation phases. Combined with previous investigations on the late Neoarchean metavolcanic rocks and granitoid orthogneisses in the northwestern part of the Eastern Hebei–Western Liaoning Province, the magmatic precursors of these monzogranitic–syenogranitic gneisses were most likely formed in a back-arc basin along the north of the Eastern Block, North China Craton.

Solution of the Ree group problem. A lecture at Rutgers University in 1979

Solution of the Ree group problem. A lecture at Rutgers University in 1979

The Existence of Minimal Logarithmic Signatures for Some Finite Simple Groups

A logarithmic signature for a finite group G is a sequence α = [A1, …, As] of subsets of G such that every element g ∈ G can be uniquely written in the form g = g1…gs, where gi ∈ Ai, 1 ⩽ i ⩽ s. The number ∑si = 1|Ai| is called the length of α and denoted by l(α). A logarithmic signature α is said to be minimal (MLS) if l(α) = ∑ni = 1mipi, where is the prime factorization of |G|. The MLS conjecture states that every finite simple group has an MLS. The aim of this article is proving the existence of a minimal logarithmic signature for the untwisted groups G2(3n), the orthogonal groups Ω7(q) and PΩ+8(q), q is an odd prime power, the orthogonal groups Ω9(3), PΩ+10(3), and PΩ−8(3), the Tits simple group 2F4(2)′, the Janko group J3, the twisted group 3D4(2), the Rudvalis group Ru, and the Fischer group Fi22. As a consequence of our results, it is proved that all finite groups of order ⩽ 1012 other than the Ree group Ree(27), the O’Nan group O′N, and the untwisted group G2(7) have MLS.

Generalised Polygons Admitting a Point-Primitive Almost Simple Group of Suzuki or Ree Type

Let $G$ be a collineation group of a thick finite generalised hexagon or generalised octagon $\Gamma$. If $G$ acts primitively on the points of $\Gamma$, then a recent result of Bamberg et al. shows that $G$ must be an almost simple group of Lie type. We show that, furthermore, the minimal normal subgroup $S$ of $G$ cannot be a Suzuki group or a Ree group of type $^2\mathrm{G}_2$, and that if $S$ is a Ree group of type $^2\mathrm{F}_4$, then $\Gamma$ is (up to point-line duality) the classical Ree-Tits generalised octagon.

Torsion units for a Ree group, Tits group and a Steinberg triality group

We investigate the Zassenhaus conjecture for the Steinberg triality group \({}^3D_4(2^3)\), Tits group \({}^2F_4(2)'\) and the Ree group \({}^2F_4(2)\). Consequently, we prove that the Prime Graph question is true for all three groups.

Recognition of finite exceptional groups of Lie type

Let q be a prime power and let G be an absolutely irreducible subgroup of GLd(F ), where F is a finite field of the same characteristic as Fq, the field of q elements. Assume that G ∼= G(q), a quasisimple group of exceptional Lie type over Fq which is neither a Suzuki nor a Ree group. We present a Las Vegas algorithm that constructs an isomorphism from G to the standard copy of G(q). If G 6∼= 3D4(q) with q even, then the algorithm runs in polynomial time, subject to the existence of a discrete log oracle.

On the uniqueness of the generalized octagon of order [formula omitted

The smallest known thick generalized octagon has order (2,4) and can be constructed from the parabolic subgroups of the Ree group F42(2). It is not known whether this generalized octagon is unique up to isomorphism. We show that it is unique up to isomorphism among those having a point a whose stabilizer in the automorphism group both fixes setwise every line on a and contains a subgroup that is regular on the set of 1024 points at maximal distance to a. Our proof uses extensively the classification of the groups of order dividing 29.

Some designs and binary codes preserved by the simple group Ru of Rudvalis

The binary codes Cn (where n∈{1755,1756,2304,2305,4059}), defined by the rank-3 primitive permutation representation of degree 4060 of the simple group Ru of Rudvalis on the cosets of the Ree group 2F4(2) are examined. These codes are obtained from the row span of the incidence matrices of the neighbourhood designs of graphs Γ, ΓR, Γ˜, Γ˜R, and ΓS defined by the union of the orbits of 2F4(2). We prove that dim(C1756)=29, dim(C2304)=28, C1756⊃C2304 and Ru acts irreducibly on C2304. Furthermore we have C1755=C2305=C4059=V4060(F2), Aut(D1755)=Aut(D1756)=Aut(D2304)=Aut(D2305)=Aut(C1756)=Aut(C2304)=Ru while Aut(D4059)=Aut(C1755)=Aut(C2305)=Aut(C4059)=S4060. We also determine the weight distribution of C1756 and C2304 and that of their duals. For each word wl of weight l, in the codes CΓ1756 or C2304 we determine the stabilizer (Ru)wl, and some primitive designs held by particular codewords.

The simple Ree groups [formula omitted] are determined by the set of their character degrees

Let G be a finite group. Let cd ( G ) be the set of all complex irreducible character degrees of G. In this paper, we will show that if cd ( G ) = cd ( H ) , where H is the simple Ree group F 4 2 ( q 2 ) , q 2 ⩾ 8 , then G ≅ H × A , where A is an abelian group. This verifies Huppertʼs Conjecture for the simple Ree groups F 4 2 ( q 2 ) when q 2 ⩾ 8 .

The norm of a Ree group

AbstractWe give an explicit construction of the Ree groups of typeG2as groups acting on mixed Moufang hexagons together with detailed proofs of the basic properties of these groups contained in the two fundamental papers of Tits on this subject (see [7] and [8]). We also give a short proof that the norm of a Ree group is anisotropic.

The norm of a Ree group

AbstractWe give an explicit construction of the Ree groups of type G2 as groups acting on mixed Moufang hexagons together with detailed proofs of the basic properties of these groups contained in the two fundamental papers of Tits on this subject (see [7] and [8]). We also give a short proof that the norm of a Ree group is anisotropic.

An experimental study of the solubility and speciation of the Rare Earth Elements (III) in fluoride- and chloride-bearing aqueous solutions at temperatures up to 300 °C

The solubility of REE(III) fluoride solids was determined in fluoride- and chloride-bearing solutions at 150, 200 and 250 °C and saturated water vapor pressure. These experimental data, together with experimental data from previously published studies, were used to evaluate formation constants for chloride- and fluoride-bearing aqueous species of the entire REE(III) group at temperatures up to 300 °C. The data show that the stability of these species differs significantly from that predicted theoretically. For example, contrary to the theoretical predictions, LREEF 2+ species are more stable than HREEF 2+ species at elevated temperature. The behavior of the chloride-bearing species is similar. Parameters for the Helgeson–Kirkham–Flowers (HKF) equation of state were determined for REEF 2+, REECl 2+ and REECl 2 + complexes using these experimental data and permit calculation of formation constants of these species at conditions not investigated experimentally. These data now permit the mobility of all REE in fluoride- and chloride-bearing hydrothermal systems to be reliably evaluated at intermediate temperatures and pressures.

On the Ree Unital

The embedding of a Ree Unital $${\mathcal{R}}(q)$$ in a finite projective plane ? of order up to q 4 is investigated when the Ree group is induced on $${\mathcal{R}}(q)$$ by a collineation group of ?. In particular, it is shown that such a embedding is not admissible for q ? 3, extending in this way a result of Luneburg dating back to 1965.

On the extension of $\mathrm {G}_{2}(3^{2n+1})$ by the exceptional graph automorphism

The main aim of this paper is to compute the character table of $\mathrm{G}_{2}(3^{2n+1})\rtimes\langle \sigma \rangle$, where $\sigma$ is the graph automorphism of $\mathrm{G}_{2}(3^{2n+1})$ such that the fixed-point subgroup $\mathrm{G}_{2}(3^{2n+1})^{\sigma}$ is the Ree group of type $\mathrm{G}_{2}$. As a consequence we explicitly construct a perfect isometry between the principal $p$-blocks of $\mathrm{G}_{2}(3^{2n+1})^{\sigma}$ and $\mathrm{G}_{2}(3^{2n+1})\rtimes\langle \sigma \rangle$ for prime numbers dividing $q^2-q+1$.

Characters of some finite groups of Lie type with a restriction containing a linear character once

Let G be a finite group of characteristic p of one of the types symplectic group Sp ( 4 , 2 f ) , Chevalley group G 2 ( q ) , Suzuki group Sz ( q ) or Ree group Re ( q ) . In this paper we show that if H is a Sylow p-subgroup of G then the restriction χ H of χ to H has at least one linear constituent of multiplicity one for certain irreducible characters χ of G.

Almost simple groups with socle Ree(q) acting on finite linear spaces

This article is part of a project set up to classify groups and linear spaces where the group acts transitively on the lines of the space. Let G be an automorphism group of a linear space. We know that the study of line-transitive finite linear spaces can be reduced to three cases, distinguishable by means of properties of the action on the point-set: that in which G is of affine type in the sense that it has an elementary abelian transitive normal subgroup; that in which G has an intransitive minimal normal subgroup; and that in which G is almost simple, in the sense that there is a simple transitive normal subgroup T in G whose centraliser is trivial, so that T ⊴ G ≤ Aut ( T ) . In this paper we treat almost simple groups in which T is a Ree group and obtain the following theorem: Let T ⊴ G ≤ Aut ( T ) , and let S be a finite linear space on which G acts as a line-transitive automorphism group. If T is isomorphic to G 2 2 ( q ) , then T is line-transitive and S is a Ree unitary space.

Block Primitive 2-(v,k,1) Designs Admitting a Ree Group of Characteristic Two

Let $$\mathcal{D}$$ be a 2-(v,k,1) design, and let G be a group of automorphisms of $$\mathcal D$$ . We show that if G is block primitive, then G does not admit a Ree group $$^2F_4(2^{2n+1})$$ as its socle.

Algebraic structure of the perfect Ree-Tits generalized octagons

We provide an algebraic description of the perfect Ree-Tits generalized octagons, i.e., an explicit embedding of octagons of this type in a 25-dimensional projective space. The construction is derived from the interplay between the 52-dimensional module of the Chevalley algebra of type F4 over a field of even characteristic and its 26-dimensional submodule. We define a quadratic duality operator that interchanges special sets of (totally) isotropic elements in those modules and establish the points of the octagon as absolute points of this duality. We introduce many algebraic operations that can be used in the study of the generalized octagon. We also prove that the Ree group acts as expected on points and pairs of points.

Intersection pattern of Ree subgroups in G2(q), q = 32n+l > 3

Abstract Throughout this paper, we fix a finite group X isomorphic to G2(q),q = 32n+1 > 3 and a subgroup G of X isomorphic to the Ree group 2G2(q). It is well-known that X contains a unique conjugacy class C of maximal subgroups with the property that each member of C is isomorphic to G ([7], Theorem A, p.33) and is the set-stabilizer in X of a Ree-Tits ovoid of the (q, q) - generalized hexagon H associated with X for the natural action of X on the points of H ([12], see also [4]). In this paper, we study the permutation representation of X on the set of the Ree-Tits ovoids of H.

The Alperin and Dade Conjectures for Ree Groups2F4(q2) in Non-defining Characteristics

This paper is part of a program to study Alperin's weight conjecture and Dade's ordinary conjecture on counting characters in blocks for several finite groups. The local structures of radical subgroups and certain radical 3-chains of a Ree group of typeF4are given and the conjectures and a conjecture of Olsson are confirmed for the group when the characteristic of the modular representation is distinct from the defining characteristic of the group.