Fischer Group Research Articles

Sporadic and Exceptional

We study the web of correspondences linking the exceptional Lie algebras E8,7,6 and the sporadic simple groups Monster, Baby and the largest Fischer group. This is done via the investigation of classical enumerative problems on del Pezzo surfaces in relation to the cusps of certain subgroups of PSL(2,R) for the relevant McKay-Thompson series in Generalized Moonshine. We also study Conway's sporadic group, as well as its association with the Horrocks-Mumford bundle.

The string C-group representations of the Suzuki, Rudvalis and O'Nan sporadic groups

We present new algorithms to classify all string C-group representations of a given group G. We use these algorithms to classify all string C-group representations of the Suzuki, Rudvalis and O’Nan sporadic groups. The new rank three algorithms also permit us to get all string C-group representations of rank three for the Conway group Co2 and the Fischer group Fi22.

On construction of certain Fischer embedded subgroups generated by 3-transpositions in Fi22

In this paper, we investigate the Fischer group [Formula: see text]. This group is generated by a conjugacy class of involutions, any non-commuting pair of which has product of order 3. Such involutions are called transpositions and their conjugacy class is denoted by D. Subgroups generated by elements of D are called D-groups as they have been called by Enright [G. M. Enright, The structure and subgroups of the Fischer groups [Formula: see text] and [Formula: see text], Ph.D. thesis, University of Cambridge (1976)], Fischer embedded or 3-transposition groups. Here, we obtain the following main results: • The rank of the groups [Formula: see text], the Weyl group W of type [Formula: see text] over fields of characteristic 2, the group M of shape [Formula: see text] and the group [Formula: see text] are computed. If G is a 3-transposition group (Fischer embedded) for a class [Formula: see text] of transpositions, then [Formula: see text] is defined as [Formula: see text], all elements in X commute[Formula: see text]. • We prove that the subgroups [Formula: see text] and [Formula: see text] are Fischer embedded and we give an explicit construction for them. It is remarkable to mention that Enright studied certain D-groups under a very strong condition that is [Formula: see text] and [Formula: see text] where G is the group generated by a proper subset E of D, [Formula: see text] is the derived subgroup of G and E is a single conjugacy class in G. Our study will be carried out without such conditions.

English

Using Curtis’s MOG [3], we display the orbits and orbit representatives for various subgroups of the Mathieu group acting on the octads of the Steiner system This information is deployed in [8] and [9] to study a graph associated with the largest simple Fischer group.

English

The disc structure of the point-line collinearity graph for the maximal 2-local geometry associated with the largest simple Fischer group is investigated. For an arbitrary vertex of this graph the first three discs are determined. Additionally a fragment of the fourth disc is uncovered.

Words for maximal Subgroups of Fi24‘

AbstractGroup Theory is the mathematical application of symmetry to an object to obtain knowledge of its physical properties. The symmetry of a molecule provides us with the various information, such as - orbitals energy levels, orbitals symmetries, type of transitions than can occur between energy levels, even bond order, all that without rigorous calculations. The fact that so many important physical aspects can be derived from symmetry is a very profound statement and this is what makes group theory so powerful. In group theory, a finite group is a mathematical group with a finite number of elements. A group is a set of elements together with an operation which associates, to each ordered pair of elements, an element of the set. In the case of a finite group, the set is finite. The Fischer groups Fi22, Fi23 and Fi24‘ are introduced by Bernd Fischer and there are 25 maximal subgroups of Fi24‘. It is an open problem to find the generators of maximal subgroups of Fi24‘. In this paper we provide the generators of 10 maximal subgroups of Fi24‘.

Monster anatomy

We investigate the two-dimensional conformal field theories (CFTs) of c=frac{47}{2} , c=frac{116}{5} and c = 23 ‘dual’ to the critical Ising model, the three state Potts model and the tensor product of two Ising models, respectively. We argue that these CFTs exhibit moonshines for the double covering of the baby Monster group, 2;cdotp;mathbb{B} , the triple covering of the largest Fischer group, 3 · Fi24′ and multiple-covering of the second largest Conway group, 2 · 21+22 · Co2. Various twined characters are shown to satisfy generalized bilinear relations involving Mckay-Thompson series. We also rediscover that the ‘self-dual’ two-dimensional bosonic conformal field theory of c = 12 has the Conway group Co0 ≃ 2 · Co1 as an automorphism group.

The number of linear factors of supersingular polynomials and sporadic simple groups

The set of prime numbers p such that the supersingular j-invariants in characteristic p are all contained in the prime field is finite. And it is well known that this set of primes coincides with the set of prime divisors of the order of the Monster simple group. In this paper, we will present analogous coincidences of supersingular invariants in level 2 and 3 and the orders of the Baby monster group and the Fischer's group. The proof uses a connection between the number of supersingular invariants and class numbers of imaginary quadratic fields.

The Computation of the 3-Modular Characters of the Fischer Group Fi23*

ABSTRACTWe determine the 35 irreducible 3-modular characters of the Fischer group Fi23. This completes the calculation of all modular character tables of this group.

Aneurysmal Subarachnoid Hemorrhage in the Elderly over Age 75: A Systematic Review.

The number of elderly patients with an aneurysmal subarachnoid hemorrhage (aSAH) has been increasing in aging- or aged societies in many countries. A treatment strategy for the elderly with aSAH has not been established, although many studies have been published emphasizing poor outcome for aSAH. The aim of this study was to analyze the factors and treatments affecting outcome in aSAH in the elderly in a systematic review of the literature by investigating patients over age 75. A literature search was done for “elderly aSAH” in PubMed and Embase. Literature with a clear description of treatment measures for aneurysmal occlusion and outcome was selected. Twelve studies, consisted of 816 cases, met the eligibility criteria. Patient characteristics included 83.2% female, 33.8% poor clinical grade on admission, 57.1% Fischer group 3, and 41% internal carotid artery aneurysm. As complications, symptomatic vasospasm was seen in 25.5% of patients, hydrocephalus in 31.1%, and medical complication in 38.4%. Favorable outcome was 35.0% in total, 45.3% for clipping, 36.3% for coiling, and 9.0% for conservative treatment. Several studies by multivariate analysis indicated that poor clinical grade on admission could be a risk factor for neurological outcome and mortality. Advanced age and selection of conservative treatment without aneurysmal occlusion could be a risk factor for mortality. Patients under age 85 with good clinical grade on admission can be candidates for treatment of aneurysm repair. However, treatment for patients over age 85 or with poor clinical grade should be carefully determined.

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.

FISCHER FIXING SYSTEMS: MOVING FORWARD WITH THE WORKFORCE - CHANGE COMMUNICATION AT THE GLOBAL DISTRIBUTION CENTER

The investment decision was made last year, and after a time-consuming selection process and intensive negotiations, the contract with the general contractor was signed. The business segment Fixing systems of the fischer group of companies faced a significant investment in its logistics – the development of automation technology in the Global Distribution Center (GDC). According to the project schedule, the transition to the automated small parts storage with shuttle technology would take place in December of this year. It was mid-April, and therefore the right time for the Head of Logistics of the business segment fischer Fixing systems, Matthias Wehle, to describe the current status: Were all the important milestones achieved and would the subsequent steps ensure a smooth transition? The employees in the logistics got accustomed to changes over the years. Structural changes such as the introduction of SAP R/3 in early 2000 and the establishment of the automated high-bay warehouse in 2008 had been successfully implemented. It was the same with the continuous improvement process through the use of the fischerProzessSystem fPS. The introduction of pick-by-light technology for the picking of small sized products with high demand had worked well, too. But the last staff meeting about launch of the shuttle system had left with him a feeling of uncertainty. He could already feel the spirit of optimism from some of the employees triggered by the upcoming introduction of the new system. The majority, however, listened patiently and Mr. Wehle sensed insecurity among the employees – how does this change affect me? Up until now, management had used multiple channels to communicate information to the employees. But feedback from the workforce to those in charge was rather rare. And the logistics manager remembered that the use of pick-by-voice for picking had been stopped for a number of reasons. Now as a result of the largest investment in recent years, the GDC faced a drastic operational change. With this in mind, Mr. Wehle pondered over the issue of employee communication again and again. And they had eight more months in order to take action until December…

Structure of the maximal -local geometry point-line collinearity graph

The point-line collinearity graph${\mathcal{G}}$of the maximal 2-local geometry for the largest simple Fischer group,$Fi_{24}^{\prime }$, is extensively analysed. For an arbitrary vertex$a$of${\mathcal{G}}$, the$i\text{th}$-disc of$a$is described in detail. As a consequence, it follows that${\mathcal{G}}$has diameter$5$. The collapsed adjacency matrix of${\mathcal{G}}$is given as well as accompanying computer files which contain a wealth of data about${\mathcal{G}}$.Supplementary materials are available with this article.

On the ranks of Fi22

If G is a finite group and X a conjugacy class of elements of G, then we define rank(G:X) to be the minimum number of elements of X generating G. Moori investigated rank(Fi22:X), where X is a conjugacy class of involutions in the Fischer group Fi22. In the present article, we completely determine the ranks of the Fischer group Fi22 by computing rank(Fi22:X), where X is any conjugacy class of Fi22.

McKay’s E 6 Observation on the Largest Fischer Group

In this paper, we study McKay’s E6-observation on the largest Fischer 3-transposition group Fi24. We investigate a vertex operator algebra \({VF^{\natural}}\) of central charge \({23\frac{1}{5}}\) on which the Fischer group Fi24 naturally acts. We show that there is a natural correspondence between dihedral subgroups of Fi24 and certain vertex operator subalgebras constructed by the nodes of the affine E6 diagram by investigating so-called derived Virasoro vectors of central charge 6/7. This allows us to reinterpret McKay’s E6-observation via the theory of vertex operator algebras.

Extensions of Groups by Weighted Steiner Loops

Solving functional equations given in Nagy and Strambach (Czechoslov. Math. J. 58(133):759–786, 2008) for extensions of a group A by a weighted Steiner loop S we obtain concrete description for all loops with interesting weak associativity properties if the Steiner loop S induces only the trivial automorphism on A. We show that the restricted Fischer groups and their geometry play an important role for loop extension with right alternative property. Also the automorphism groups of these extensions as well as the conditions for isomorphisms between two extensions are studied.

Pointed Hopf algebras over the sporadic simple groups

We show that every finite-dimensional complex pointed Hopf algebra with group of group-likes isomorphic to a sporadic group is a group algebra, except for the Fischer group Fi 22 , the Baby Monster and the Monster. For these three groups, we give a short list of irreducible Yetter–Drinfeld modules whose Nichols algebra is not known to be finite-dimensional.

A 3-local characterization of [formula omitted

We identify Fischer's group, Fi 24 ′ , by the structure of the centralizer of a 3-central element. Also as a corollary we will identify this group from two of its 3-locals.

The Fischer-Clifford Matrices and Character Table of a Maximal Subgroup of Fi24

The Fischer group [Formula: see text] is the largest 3-transposition sporadic group of order 2510411418381323442585600 = 222.316.52.73.11.13.17.23.29. It is generated by a conjugacy class of 306936 transpositions. Wilson [15] completely determined all the maximal 3-local subgroups of Fi24. In the present paper, we determine the Fischer-Clifford matrices and hence compute the character table of the non-split extension 37· (O7(3):2), which is a maximal 3-local subgroup of the automorphism group Fi24 of index 125168046080 using the technique of Fischer-Clifford matrices. Most of the calculations are carried out using the computer algebra systems GAP and MAGMA.

Pointed Hopf algebras over some sporadic simple groups

Any finite-dimensional complex pointed Hopf algebra with group of group-likes isomorphic to a sporadic group, with the possible exception of the Fischer group Fi 22 , the Baby Monster B and the Monster M, is a group algebra.