Group Sym Research Articles

Matsuo algebras in characteristic 2

We extend the theory of Matsuo algebras, which are certain non-associative algebras related to 3-transposition groups, to characteristic 2. The decompositions of our algebras are now induced by nilpotent elements associated to lines in the corresponding Fischer space, rather than idempotent elements associated to points. For many 3-transposition groups, this still gives rise to a Z/2Z-graded fusion law, and we provide a complete classification of when this occurs. In one particular small case, arising from the 3-transposition group Sym(4), the fusion law is even stronger, and the resulting Miyamoto group is an algebraic group Ga2⋊Gm.

On the intersection density of the symmetric group acting on uniform subsets of small size

Given a finite transitive group G≤Sym(Ω), a subset F of G is intersecting if any two elements of F agree on some elements of Ω. The intersection density of G, denoted by ρ(G), is the maximum of the rational number |F|(|G||Ω|)−1 when F runs through all intersecting sets in G. In this paper, we prove that if G is the group Sym(n) or Alt(n) acting on the k-subsets of {1,2,3…,n}, for k∈{3,4,5}, then ρ(G)=1. Our proof relies on the representation theory of the symmetric group and the ratio bound.

Effective dose of intravenous oxycodone depending on sex and age for attenuation of intubation-related hemodynamic responses

Background/aimPreoperative intravenous oxycodone may help to prevent or attenuate intubation-related hemodynamic responses (IRHRs), but its pharmacokinetics differs according to age and sex. Therefore, we investigated the 95% effective dose (ED95) of intravenous oxycodone for attenuating all IRHRs, depending on the age and sex of the study population. Materials and methodsAll patients were allocated to one of 6 groups: 1) 20–40 year old males, 2) 41–65yearold males, 3) 66–80 year old males, 4) 20–40 year old females, 5) 41–65yearold females, and 6) 66–80 year old females (groups YM, OM, EM, YF, OF, and EF, respectively). Using Dixon’s up-and-down method, the first patient in each group was slowly injected with intravenous oxycodone (0.1 mg kg-1) 20 min before intubation. The subsequent patient received the next oxycodone dose, which was decreased or increased by 0.01 mg kg-1, depending on the “success” or “failure” of attenuation of all IRHRs to within 20% of the baseline values at 1 min after intubation in the previous patient. After obtaining 8 crossover points, predictive ED95 was estimated with probit regression analysis.ResultsED95 varied greatly according to age and sex. ED95was 0.133 mg kg-1, 0.181 mg kg-1, 0.332 mg kg-1, 0.183 mg kg-1, 0.108 mg kg-1, and 0.147 mg kg-1in groups YM, OM, EM, YF, OF, and EF, respectively.ConclusionED95 is higher in males with increasing age but is ambiguous for females. ED95 is higher in males than in females over 40 years of age but is higher in females than in males under 41 years of age. However, after considering the age and sex of the study population, these results can be used as reference doses for further studies to verify the clinical effects of oxycodone for attenuating all IRHRs.

Cellularity of endomorphism algebras of Young permutation modules

Let E be an n-dimensional vector space. Then the symmetric group Sym(n) acts on E by permuting the elements of a basis and hence on the r-fold tensor product E⊗r. Bowman, Doty and Martin ask, in [1], whether the endomorphism algebra EndSym(n)(E⊗r) is cellular. The module E⊗r is the permutation module for a certain Young Sym(n)-set. We shall show that the endomorphism algebra of the permutation module on an arbitrary Young Sym(n)-set is a cellular algebra. We determine, in terms of the point stabilisers which appear, when the endomorphism algebra is quasi-hereditary.

Commuting Graphs, C(G, X) in Symmetric Groups Sym(n) and Its Connectivity

A commuting graph is a graph denoted by C ( G , X ) where G is any group and X, a subset of a group G, is a set of vertices for C ( G , X ) . Two distinct vertices, x , y ∈ X , will be connected by an edge if the commutativity property is satisfied or x y = y x . This study presents results for the connectivity of C ( G , X ) when G is a symmetric group of degree n, Sym ( n ) , and X is a conjugacy class of elements of order three in G.

Equivariant dissipation in non-archimedean groups

We prove that, if a topological group G has an open subgroup of infinite index, then every net of tight Borel probability measures on G UEB-converging to invariance dissipates in G in the sense of Gromov. In particular, this solves a 2006 problem by Pestov: for every left-invariant (or right-invariant) metric d on the infinite symmetric group Sym(ℕ), compatible with the topology of pointwise convergence, the sequence of the finite symmetric groups (Sym(n), d ↾Sym(n), μSym(n))n∈ℕ equipped with the restricted metrics and their normalized counting measures dissipates, thus fails to admit a subsequence being Cauchy with respect to Gromov’s observable distance.

The 2-Closure of a $${\textstyle{3 \over 2}}$$ 3 2 -Transitive Group in Polynomial Time

Let G be a permutation group on a finite set Ω. The k-closure G(k) of G is the largest subgroup of the symmetric group Sym(Ω) having the same orbits with G on the kth Cartesian power Ωk of Ω. The group G is called $${\textstyle{3 \over 2}}$$ -transitive, if G is transitive and the orbits of a point stabilizer Gα on Ω{α} are of the same size greater than 1. We prove that the 2-closure G(2) of a $${\textstyle{3 \over 2}}$$ -transitive permutation group G can be found in polynomial time in size of Ω. Moreover, if the group G is not 2-transitive, then for every positive integer k its k-closure can be found within the same time. Applying the result, we prove the existence of a polynomial-time algorithm for solving the isomorphism problem for schurian $${\textstyle{3 \over 2}}$$ -homogeneous coherent configurations, that is coherent configurations naturally associated with $${\textstyle{3 \over 2}}$$ -transitive groups.

Cherlin's conjecture for almost simple groups of Lie rank 1

AbstractA permutation group G on a set Ω is said to be binary if, for every n ∈ ℕ and for every I, J ∈ Ωn, the n-tuples I and J are in the same G-orbit if and only if every pair of entries from I is in the same G-orbit to the corresponding pair from J. This notion arises from the investigation of the relational complexity of finite homogeneous structures.Cherlin has conjectured that the only finite primitive binary permutation groups are the symmetric groups Sym(n) with their natural action, the groups of prime order, and the affine groups V ⋊ O(V) where V is a vector space endowed with an anisotropic quadratic form.We prove Cherlin's conjecture, concerning binary primitive permutation groups, for those groups with socle isomorphic to PSL2(q), 2B2(q), 2G2(q) or PSU3(q). Our method uses the notion of a “strongly non-binary action”.

Minimal generation of transitive permutation groups

This paper discusses upper bounds on the minimal number of elements d(G) required to generate a transitive permutation group G, in terms of its degree n, and its order |G|. In particular, we reduce a conjecture of L. Pyber on the number of subgroups of the symmetric group Sym(n). We also prove that our bounds are best possible.

Multicenter epidemiological investigation of hospitalized elderly, young and middle-aged patients with severe burn

Multicenter epidemiological investigation of hospitalized elderly, young and middle-aged patients with severe burn

Effects of Meek skin grafting on patients with extensive deep burn at different age groups

Objective: To investigate the effect of Meek skin grafting on patients with extensive deep burn at different age groups. Methods: Eighty-four patients with extensive deep burns conforming to the study criteria were hospitalized in our unit from April 2011 to April 2015. Patients were divided into children group (C, with age less than 12 years old), young and middle-aged group (YM, with age more than 18 years and less than 50 years old), and old age group (O, with age more than 55 years old) according to age, with 28 patients in each group. All patients received Meek skin grafting treatment. The use of autologous skin area, operation time, wound healing time, and hospitalization time were recorded. The survival rate of skin graft on post operation day 7, complete wound healing rate in post treatment week 2, and the mortality were calculated. Data were processed with one-way analysis of variance, t test, and χ(2) test. Results: The use of autologous skin area of patients in group C was (5.1±1.0)% total body surface area (TBSA), significantly less than (8.3±1.0)%TBSA and (8.3±1.4)%TBSA in groups YM and O, respectively (with t values 32.900 and 52.624, respectively, P values below 0.05). The operation time, wound healing time, and hospitalization time of patients in group C were (1.368±0.562) h, (9.6±0.6) and (32±11) d, significantly shorter than those in group YM [(3.235±0.011) h, (16.9±2.6) and (48±12) d, respectively] and group O [(3.692±0.481) h, (17.3±2.6) and (46±13) d, respectively, with t values from 4.350 to 21.160, P values below 0.05]. The survival rate of skin graft of patients on post operation day 7 in group C was (92±15)%, significantly higher than (81±10)% and (72±12)% in groups YM and O, respectively (with t values 5.509 and 3.229, respectively, P values below 0.05). The above indexes in groups YM and O were similar (with t values from 0.576 to 22.958, P values above 0.05). Complete wound healing rate in post treatment week 2 and the mortality of patients in group C were similar to those in groups YM and O (with χ(2) values 0.365 and 0.122, respectively, P values above 0.05). Conclusions: Meek skin grafting can be used in the treatment of patients with extensive deep burns at different age groups, compared with the young and middle-aged and old patients, the effect in children was better.

The Minimal Faithful Permutation Degree for a Direct Product Obeying an Inequality Condition

The minimal faithful permutation degree μ(G) of a finite group G is the least nonnegative integer n such that G embeds in the symmetric group Sym(n). Clearly μ(G × H) ≤ μ(G) + μ(H) for all finite groups G and H. In 1975, Wright ([10]) proved that equality occurs when G and H are nilpotent and exhibited an example of strict inequality where G × H embeds in Sym(15). In 2010 Saunders ([7]) produced an infinite family of examples of permutation groups G and H where μ(G × H) < μ(G) + μ(H), including the example of Wright's as a special case. The smallest groups in Saunders’ class embed in Sym(10). In this article, we prove that 10 is minimal in the sense that μ(G × H) = μ(G) + μ(H) for all groups G and H such that μ(G × H) ≤9.

On groups generated by involutions of a semigroup

An involution on a semigroup S (or any algebra with an underlying associative binary operation) is a function α:S→S that satisfies α(xy)=α(y)α(x) and α(α(x))=x for all x,y∈S. The set I(S) of all such involutions on S generates a subgroup C(S)=〈I(S)〉 of the symmetric group Sym(S) on the set S. We investigate the groups C(S) for certain classes of semigroups S, and also consider the question of which groups are isomorphic to C(S) for a suitable semigroup S.

Semidefinite programming for permutation codes

We initiate study of the Terwilliger algebra and related semidefinite programming techniques for the conjugacy scheme of the symmetric group Sym(n). In particular, we compute orbits of ordered pairs on Sym(n) acted upon by conjugation and inversion, explore a block diagonalization of the associated algebra, and obtain improved upper bounds on the size M(n,D) of permutation codes of lengths n=6,7. For instance, these techniques detect the nonexistence of the projective plane of order six via M(6,5)<30 and yield a new upper bound M(7,4)≤535 for a challenging open case. Each of these represents an improvement on earlier Delsarte linear programming results.

On Pairs of Commuting Involutions in Sym(n) and Alt(n)

The number of pairs of commuting involutions in Sym(n) and Alt(n) is determined up to isomorphism. It is also proven that, up to isomorphism and duality, there are exactly two abstract regular polyhedra on which the group Sym(6) acts as a regular automorphism group.

Normal coverings and pairwise generation of finite alternating and symmetric groups

The normal covering number γ(G) of a finite, non-cyclic group G is the least number of proper subgroups such that each element of G lies in some conjugate of one of these subgroups. We prove that there is a positive constant c such that, for G a symmetric group Sym(n) or an alternating group Alt(n), γ(G)⩾cn. This improves results of the first two authors who had earlier proved that aφ(n)⩽γ(G)⩽2n/3, for some positive constant a, where φ is the Euler totient function. Bounds are also obtained for the maximum size κ(G) of a set X of conjugacy classes of G=Sym(n) or Alt(n) such that any pair of elements from distinct classes in X generates G, namely cn⩽κ(G)⩽2n/3.

A note on the symmetry group of full rank perfect binary codes

It is proved that the size of the symmetry group Sym(C) of every full rank perfect 1-error correcting binary code C of length n is less than or equal to 2|Sym(Hn)|/(n+1), where Hn is a Hamming code of the same length.

On the length of chains of proper subgroups covering a topological group

We prove that if an ultrafilter L is not coherent to a Q-point, then each analytic non-sigma-bounded topological group G admits an increasing chain <G_a : a < b(L)> of its proper subgroups such that: (i) U_{a in b(L)} G_a=G; and $(ii)$ For every sigma-bounded subgroup H of G there exists a such that H is a subset of G_a. In case of the group Sym(w) of all permutations of w with the topology inherited from w^w this improves upon earlier results of S. Thomas.

Finitely presented algebras and groups defined by permutation relations

The class of finitely presented algebras over a field K with a set of generators a 1 , … , a n and defined by homogeneous relations of the form a 1 a 2 ⋯ a n = a σ ( 1 ) a σ ( 2 ) ⋯ a σ ( n ) , where σ runs through a subset H of the symmetric group Sym n of degree n , is introduced. The emphasis is on the case of a cyclic subgroup H of Sym n of order n . A normal form of elements of the algebra is obtained. It is shown that the underlying monoid, defined by the same (monoid) presentation, has a group of fractions and this group is described. Properties of the algebra are derived. In particular, it follows that the algebra is a semiprimitive domain. Problems concerning the groups and algebras defined by arbitrary subgroups H of Sym n are proposed.

Enumerating Groups Acting Regularly on a d -Dimensional Cube

In the survey article [2] it was noted, among many other open problems, that the classification of the groups acting regularly on a d-dimensional cube Γ is unsettled. In other words, the classification of the finite groups G such that Cay(G, S) ≅ Γ, for some subset S of G, is still unknown. In this article, we prove that there are at least 2 d 2/64 − (d/2)log2(d/2) nonisomorphic 2-groups of Frattini class 2 acting regularly on a d-dimensional cube. Other relevant results are presented. As a corollary of our result, we remark that the symmetric group Sym(n) on n symbols contains at least 2 n 2/256 − (n/4)log2(n/4) subgroups up to isomorphism. In particular, we recall that in [4] it was proved that the total number of subgroups of Sym(n) is at most 2 cn 2 , for c = log224.