On geometries of the Conway group Co 3 and their McLaughlin group subgeometries

In 1970 Bernd Fischer proved his beautiful theorem classifying the almost simple groups generated by 3-transpositions, and in the process discovered three new sporadic groups, now known as the Fischer groups. Since then, the theory of 3-transposition groups has become an important part of finite simple group theory, but Fischer's work has remained unpublished. 3-Transposition Groups contains the first published proof of Fischer's Theorem, written out completely in one place. Fischer's result, while important and deep (covering a number of complex examples), can be understood by any student with some knowledge of elementary group theory and finite geometry. Thus Part I has minimal prerequisites and could be used as a text for an intermediate level graduate course. Parts II and III are aimed at specialists in finite groups and are a step in the author's program to supply a strong foundation for the theory of sporadic groups.

Word representation of cords on a punctured plane

In [Spectral asymmetry and Riemannian geometry. III, Math. Proc. Cambridge Philos. Soc. 79 (1976) 71–99] Atiyah, Patodi and Singer introduced spectral flow for elliptic operators on odd-dimensional compact manifolds. They argued that it could be computed from the Fredholm index of an elliptic operator on a manifold of one higher dimension. A general proof of this fact was produced by Robbin–Salamon [The spectral flow and the Maslov index, Bull. London Math. Soc. 27(1) (1995) 1–33, MR 1331677]. In [F. Gesztesy, Y. Latushkin, K. Makarov, F. Sukochev and Y. Tomilov, The index formula and the spectral shift function for relatively trace class perturbations, Adv. Math. 227(1) (2011) 319–420, MR 2782197], a start was made on extending these ideas to operators with some essential spectrum as occurs on non-compact manifolds. The new ingredient introduced there was to exploit scattering theory following the fundamental paper [A. Pushnitski, The spectral flow, the Fredholm index, and the spectral shift function, in Spectral Theory of Differential Operators, American Mathematical Society Translations: Series 2, Vol. 225 (American Mathematical Society, Providence, RI, 2008), pp. 141–155, MR 2509781]. These results do not apply to differential operators directly, only to pseudo-differential operators on manifolds, due to the restrictive assumption that spectral flow is considered between an operator and its perturbation by a relatively trace-class operator. In this paper, we extend the main results of these earlier papers to spectral flow between an operator and a perturbation satisfying a higher pth Schatten class condition for [Formula: see text], thus allowing differential operators on manifolds of any dimension [Formula: see text]. In fact our main result does not assume any ellipticity or Fredholm properties at all and proves an operator theoretic trace formula motivated by [M.-T. Benameur, A. Carey, J. Phillips, A. Rennie, F. Sukochev and K. Wojciechowski, An analytic approach to spectral flow in von Neumann algebras, in Analysis, Geometry and Topology of Elliptic Operators (World Scientific Publisher, Hackensack, NJ, 2006), pp. 297–352, MR 2246773; A. Carey, H. Grosse and J. Kaad, On a spectral flow formula for the homological index, Adv. Math. 289 (2016) 1106–1156, MR 3439708]. We illustrate our results using Dirac type operators on [Formula: see text] for arbitrary [Formula: see text] (see Sec. 8). In this setting Theorem 6.4 substantially extends Theorem 3.5 of [A. Carey, F. Gesztesy, H. Grosse, G. Levitina, D. Potapov, F. Sukochev and D. Zanin, Trace formulas for a class of non-Fredholm operators: a review, Rev. Math. Phys. 28(10) (2016) 1630002, MR 3572626], where the case d = 1 was treated.

Max Schiffer was twice honored by invitations to address the International Congress of Mathematicians, at Cambridge (Massachusetts) in 1950 and at Edinburgh in 1958. The articles (Schiffer, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, American Mathematical Society, Providence, vol. 2, pp. 233–240, 1952; Schiffer, Proceedings of the International Congress of Mathematicians, Edinburgh, 1958, Cambridge University Press, New York, pp. 211–231, 1960) are written versions of his lectures. Both of the articles are devoted to the variational methods for which Schiffer was best known, but they are rather different in character. The paper (Schiffer, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, American Mathematical Society, Providence, vol. 2, pp. 233–240, 1952) offers a broad discussion of variational methods and their relative merits, and it surveys a variety of applications. The paper (Schiffer, Proceedings of the International Congress of Mathematicians, Edinburgh, 1958, Cambridge University Press, New York, pp. 211–231, 1960), on the other hand, focuses more narrowly on applications to particular extremal problems in function theory such as coefficient problems, and it gives a fairly detailed account of technical advances in the use of variational methods. In this respect, (Schiffer, Proceedings of the International Congress of Mathematicians, Edinburgh, 1958, Cambridge University Press, New York, pp. 211–231, 1960) is somewhat dated, since the Bieberbach conjecture has now been proved [see commentaries on (Schiffer, Proc. London Math. Soc., 44(2), 432–449, 1938; Schiffer, Proc. London Math. Soc., 44(2), 450–452, 1938; Schiffer and Charzynski, Arch. Rational Mech. Anal., 5, 187–193, 1960)] and the conjecture \(\vert b_{n}\vert \leq \frac{2} {n+1}\) for functions of class Σ has been disproved for all n ≥ 3 [see commentaries on (Schiffer, Bull. Soc. Math. France, 66, 48–55, 1938; Schiffer and Garabedian, Ann. of Math., 61(2), 116–136, 1955; Schiffer et al., J. Analyse Math., 40, 203–238, 1981)]. The paper (Schiffer, Proceedings of the International Congress of Mathematicians, Edinburgh, 1958, Cambridge University Press, New York, pp. 211–231, 1960) also gives an extended description of Schiffer’s method for obtaining a lower bound for the first nontrivial Fredholm eigenvalue of a simply connected domain with analytic boundary curve. Schiffer and others returned repeatedly to this problem [see Kuhnau’s commentary on (Schiffer, Pacific J. Math., 7, 1187–1225, 1957; Schiffer, Pacific J. Math., 9, 211–269, 1959; Schiffer, Rend. Mat., 22(5), 447–468, 1963; Schiffer, Ann. Polon. Math., 39, 149–164, 1981)]. The lower bound is important numerically, because it allows an estimate on the rate at which the Neumann series converges to the solution of the classical Poincare–Fredholm integral equation associated with the solution of a Dirichlet problem. In his expository paper (Schiffer, Rend. Mat. 22(5), 447–468, 1963), Schiffer gives a clear and detailed account of this beautiful circle of ideas.

Rational rigidity and the sporadic groups

Amalgams, blocks, weights, fusion systems and finite simple groups

This is an introduction to finite simple groups, in particular sporadic groups, intended for physicists. After a short review of group theory, we enumerate the 1+1+16=18 families of finite simple groups, as an introduction to the sporadic groups. These are described next, in three levels of increasing complexity, plus the six isolated ''pariah'' groups. The (old) five Mathieu groups make up the first, smallest order level. The seven groups related to the Leech lattice, including the three Conway groups, constitute the second level. The third and highest level contains the Monster group M, plus seven other related groups. Next a brief mention is made of the remaining six pariah groups, thus completing the 5+7+8+6=26 sporadic groups. The review ends up with a brief discussion of a few of physical applications of finite groups in physics, including a couple of recent examples which use sporadic groups.

A conjecture of Frobenius which has been reduced to the classification of finite simple groups is verified for the sporadic simple groups. Let G be a finite group and n be a positive integer dividing IGI. Let Ln(G) = {x E G xn = 1). Then by a theorem of Frobenius [6] one knows that ILn(G)I = cnn for some integer cn. Frobenius conjectured that Ln(G) forms a subgroup of G provided ILn(G)I = n (see [2]). Zemlin [25] has reduced the conjecture to the classification of finite simple groups which is now complete (see [8]). The author has verified the conjecture for the Fischer Griess monster F1 and the Fischer baby monster F2 in [24]. The purpose of this note is to prove the following THEOREM. The conjecture of Frobenius is true for all the sporadic simple groups. The proof of our theorem has been carried out in the following way with the use of a computer. Let G be one of the sporadic simple groups. By [24] we may assume that G # F1 and G # F2. Let f(G, t) be the number of elements of order t in G and Ord(G) = {order of x I x E G}. Tables of f(G, t) are given in the Appendix; see the supplements section at the end of this issue. For f(G, t) the reader is referred to the following papers: Ml1, M22, M23 Burgoyne and Fong [1] M12, M24 Frobenius [5] ii Janko [14] HJ = J2 Hall and Wales [8] HJM= J3 Janko [15] J4 Janko [16] HiS Frame [4] Suz Wright [23] McL, .3 Finkelstein [3] Rud Rudvalis [19] HHM Held [10] LyS Lyons [17] Received June 21, 1982; revised March 19, 1984 and July 11, 1984. 1980 Mathematics Subject Classification. Primary 20D05.

Let Gbe a finite group and Sa sporadic simple group. We denote by π(G) the set of all primes dividing the order of G. The prime graph Γ(G) of Gis defined in the usual way connecting pand qin π(G) when there is an element of order pqin G. The main purpose of this paper is to determine finite group Gsatisfying Γ(G) = Γ(S) (See Theorem 3) and to give applications which generalize Abe (Abe, S. Two ways to characterize 26 sporadic finite simple groups. Preprint) and Chen (Chen, G. (1996). A new characterization of sporadic simple groups. Algebra Colloq.3:49–58). The results are elementary but quite useful.

Deficiency and the geometric invariants of a group (with an appendix by Pascal Schweitzer)

Let k be an algebraically closed field of prime characteristic p. Let G be a finite group, let N be a normal subgroup of G, and let c be a G-stable block of kN so that ( k ⁢ N ) ⁢ c {(kN)c} is a p-permutation G-algebra. As in Section 8.6 of [M. Linckelmann, The Block Theory of finite Group Algebras: Volume 2, London Math. Soc. Stud. Texts 92, Cambridge University, Cambridge, 2018], a ( G , N , c ) {(G,N,c)} -Brauer pair ( R , f R ) {(R,f_{R})} consists of a p-subgroup R of G and a block f R {f_{R}} of ( k ⁢ C N ⁢ ( R ) ) {(kC_{N}(R))} . If Q is a defect group of c and f Q ∈ 𝐵 ⁢ ℓ ⁡ ( k ⁢ C N ⁢ ( Q ) ) {f_{Q}\in\operatorname{\textit{B}\ell}(kC_{N}(Q))} , then ( Q , f Q ) {(Q,f_{Q})} is a ( G , N , c ) {(G,N,c)} -Brauer pair. The ( G , N , c ) {(G,N,c)} -Brauer pairs form a (finite) poset. Set H = N G ⁢ ( Q , f Q ) {H=N_{G}(Q,f_{Q})} so that ( Q , f Q ) {(Q,f_{Q})} is an ( H , C N ⁢ ( Q ) , f Q ) {(H,C_{N}(Q),f_{Q})} -Brauer pair. We extend Lemma 8.6.4 of the above book to show that if ( U , f U ) {(U,f_{U})} is a maximal ( G , N , c ) {(G,N,c)} -Brauer pair containing ( Q , f Q ) {(Q,f_{Q})} , then ( U , f U ) {(U,f_{U})} is a maximal ( H , C N ⁢ ( c ) , f Q ) {(H,C_{N}(c),f_{Q})} -Brauer pair containing ( Q , f Q ) {(Q,f_{Q})} and conversely. Our main result shows that the subcategories of ℱ ( U , f U ) ⁢ ( G , N , c ) {\mathcal{F}_{(U,f_{U})}(G,N,c)} and ℱ ( U , f U ) ⁢ ( H , C N ⁢ ( Q ) , f Q ) {\mathcal{F}_{(U,f_{U})}(H,C_{N}(Q),f_{Q})} of objects between and including ( Q , f Q ) {(Q,f_{Q})} and ( U , f U ) {(U,f_{U})} are isomorphic. We close with an application to the Clifford theory of blocks.

Let G be a finite group. Denote by Irr(G) the set of all irreducible complex characters of G. Let cd(G) be the set of all irreducible complex character degrees of G forgetting multiplicities, that is, cd(G) = {χ(1) : χ ∈ Irr(G)} and let cd*(G) be the set of all irreducible complex character degrees of G counting multiplicities. Let H be an alternating group of degree at least 5, a sporadic simple group or the Tits group. In this paper, we will show that if G is a non-abelian simple group and \(cd(G)\subseteq cd(H)\) then G must be isomorphic to H. As a consequence, we show that if G is a finite group with \(cd^*(G)\subseteq cd^*(H)\) then G is isomorphic to H. This gives a positive answer to Question 11.8 (a) in (Unsolved problems in group theory: the Kourovka notebook, 16th edn) for alternating groups, sporadic simple groups or the Tits group.

This meeting was well attended with 17 participants with broad geographic representation from 3 continents. There were 16 talks during the workshop including an invited talk by Anda Degeratu a participant in the competing String Theory workshop. The method of group amalgams is a highly effective way of classifying mathematical objects possessing high degrees of symmetry. The idea of the method is separation of the study of the local structure of the acting group from the question of it's global isomorphism type. The method of group amalgams has been successfully applied to problems in graph theory and diagram geometry. It also featured prominently in group theory. For example, the fact that the Monster sporadic simple group is the universal completion of the amalgam associated with the tilde geometry formed the foundation of the solution of the famous Y -group conjecture that the Y_{555} presentation defines the Bimonster (the direct product of two copies of the Monster sporadic simple group extended by a group of order 2). J.H. Conway coined for this theorem the name 'NICE' where \textbf{N} is for \textbf{N}orton, \textbf{I} for \textbf{I}vanov, \textbf{C} for \textbf{C}onway and \textbf{E} for anyone \textbf{E}lse. The proof of the NICE theorem based on the method of group amalgams is presented in the two volume monograph of the organisers published by Cambridge University Press. \par Recently a dramatic progress was made within the study of flag-transitive diagram geometries. The importance of the notion of \textit{constrained} completions of amalgams was realised. Within this framework many geometries of sporadic groups were characterised as the constrained completions of suitable amalgams. This approach also gives a general criterion about possible shapes of diagrams of flag-transitive geometries. This enables the area of diagram geometries to leave it's "botanical" stage of example collection and enter the stage of theory building. \par Among other applications we would like to mention recent applications of the amalgam method to the cohomologies of finite groups. These ideas were described in the notes of M. Aschbacher on calculation of the Schur multiples of some finite simple groups. \par During the workshop we had discussed in detail the proofs of a number of results obtained along the lines of the amalgam method, as well as of directions of future research. We believe that the abstract of the talks given at the workshop facilitate for the younger mathematicians access to these extremely important, yet very technically complex tools of mathematical research.

One of the famous open problems in Finite Geometry is the classification of ovoids in a projective 3-space PG(3, q) over the finite field with q elements, q even. For q odd, this classification was obtained in 1955 by A. Barlotti [ A. Barlotti, Un'estensione del teorema di Segre-Kustaanheimo. Boll. Un. Mat. Ital. (3) 10 (1955), 498–506. MR0075606 (17,776b) Zbl 0066.38901 ] and G. Panella [ G. Panella, Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito. Boll. Un. Mat. Ital. (3) 10 (1955), 507–513. MR0075607 (17,776c) Zbl 0066.38902 ]. A breakthrough result was a recent one of M. R. Brown, who showed in [ M. R. Brown, Ovoids of PG (3, q), q even, with a conic section. J. London Math. Soc. (2) 62 (2000), 569–582. MR1783645 (2001i:51012) Zbl 1038.51008 ] that when such an ovoid has a conic plane section, the ovoid must be an elliptic quadric. Even more recently, M. R. Brown and M. Lavrauw [ M. R. Brown, M. Lavrauw, Eggs in PG(4n − 1, q), q even, containing a pseudo-conic. Bull. London Math. Soc. 36 (2004), 633–639. MR2070439 (2005k:51009) Zbl 1064.51005 ] generalized this theorem by obtaining a similar result for higher dimensions. Both results have equivalent statements in the theory of (translation) generalized quadrangles. In this note, we improve these results by showing that an elation generalized quadrangle of order (q, q 2), q even, with a subGQ containing the elation point arises from a non-singular elliptic quadric in PG(5, q). The theorem itself arises as a corollary of a more general observation which works for all characteristics. There is a wealth of consequences.

Let $G$ be a finite group‎. ‎We say that $G$ has emph{spread} r if for any set of distinct non-trivial elements of $G$ $X:={x_1,ldots‎, ‎x_r}subset G^{#}$ there exists an element $yin G$ with the property that $langle x_i,yrangle=G$ for every $1leq ileq r$‎. ‎We say $G$ has emph{exact spread} $r$ if $G$ has spread $r$ but not $r+1$‎. ‎The spreads of finite simple groups and their decorations have been much-studied since the concept was first introduced by Brenner and Wiegold in the mid 1970s‎. ‎Despite this‎, ‎the exact spread of very few finite groups‎, ‎and in particular of the finite simple groups and their decorations‎, ‎is known‎. ‎Here we calculate the exact spread of the sporadic simple Mathieu group M$_{23}$‎, ‎proving that it is equal to 8064‎. ‎The precise value of the exact spread of a sporadic simple group is known in only one other case‎ - ‎the Mathieu group M$_{11}$‎.