Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups

Description

This atlas covers groups from the families of the classification of finite simple groups. Recently updated incorporating corrections

heard about on MathWorld's Dynkin Diagram entry

At last, an official collection of character tables and related information about many finite simple groups has appeared in book form. This information is important to specialists in finite group theory and the volume contains neatly presented instructional material which the nonspecialists can appreciate. For years, the authors have used the material at a very high level. It has been reworded and refined by experience. At the month-long 1979 Santa Cruz conference on finite groups, Simon Norton carried a shopping bag of tattered printouts and character tables to deal with urgent questions about simple groups. Now, we can all have the power of such rapid access, but in a classier format!
The "classic'' character table of a finite group G is by definition a k×k matrix of complex numbers, whose rows are indexed by the k irreducible characters and whose columns are indexed by the k conjugacy classes; of course, it is not unique because there is no generally accepted way to order the index sets, though the principal character (corresponding to the trivial homomorphism G→GL(1,C)) is always listed first. The (i,j) entry is χi(gj), the value of the ith irreducible character on a representative of the jth conjugacy class, and this algebraic number is always a sum of d |gj|th roots of unity, where d=χi(1) is the degree of χi.
The efforts of the last 25 years to classify finite simple groups created a greater need to have numerical and combinatorial information about the known groups. The occasional tables produced by R. Brauer or J. S. Frame or J. Todd years ago were followed by a flood of tables in the 1960s and 1970s. Generally, these were distributed informally, often with no name or source written on them and always without proof. Referring to a character table in a research article was awkward at times. The general theory of Brauer gave many arithmetic conditions on the character table which in "easy'' cases allowed one to fill in many blank entries for the table of a particular group. This was not always the case. For instance, David Hunt's work on the tables for the Fischer 3-transposition groups took an especially long time and involved extensive computer work and a study of induct-restrict tables for subgroups with known character tables.
In sum, the five authors have collected some of this early and unpublished work, then greatly extended it and put it in a form suitable for easy modern applications.
The book is organized as follows: (I) Introduction and explanations (28 pages), (II) The character tables (235 pages), (III) Supplementary tables (6 pages), (IV) References (8 pages) and Index (1 page).
(I): Sections 1, 2 and 3 contain a rapid introduction to the families of finite simple groups. It is clear and telegraphic in style and not intended for someone who is looking for full discussions and constructions.
Sections 4 through 7 discuss the multiplier, automorphism groups, isoclinism and the group extension theory which is relevant to interpreting the blocks (and broken-edge blocks) in the tables, notation for conjugacy classes, algebraic numbers and algebraic conjugates of these two concepts. We comment on the tables themselves in (II). The authors' notations for algebraic integers are very successful for character tables, e.g., z=zN=exp(2πi/N), bN=12∑N−1t=1zt2, cN=13∑N−1t=1zt3 (for N≡1 (mod 3)), etc.
One fault with the exposition is that the authors use terms and notation without explanation, then define them later. In the above sequence of definitions, for zN, bN, cN,⋯, one finds "n2'', but not a definition until further down the column. The notation ∗k is used in Section 7.3 but no hint is given for where to look for the definition. It would help if an index of notations and definitions were included to help the reader who starts reading in the middle.
The authors discuss the several existing systems of notation for the simple groups. Parts of the system used in the Atlas make the reviewer uncomfortable.
The most glaring item is the use of "O'' for the simple composition factor of the n-dimensional orthogonal group of type ϵ over Fq. In other systems, this group would be PΩϵ(n,q) or one of Dm(q), 2Dm(q) (when n=2m) or Bm(q) when n=2m+1. The authors reject these notations because they want one letter for the basic name of all these simple groups.
The second comment is about names assigned to sporadic groups; see Table 1, page viii. The principle generally used by group theorists has been to name a sporadic group after its discoverers and use a symbol related to these names. The sometime exceptions to this have been the Conway groups (denoted by .0,.1,.2 and .3 since 1968 but by Co0,Co1,Co2, and Co3 in this volume), the Fischer groups (denoted by M(22),M(23) and M(24)′ originally, but later by Fi22,Fi23 and Fi′24) and the Monster (the group discovered by Fischer and the reviewer in November 1973; the Atlas symbols are M, FG and F1) and the Baby Monster (the {3,4}+-transposition group discovered by Fischer earlier in 1973; the Atlas symbols are B and F2) and the Harada group (called the Harada-Norton group in the Atlas; the Atlas symbols are HN and F5).
The system of F's with subscripts has several nice group-theoretic features. However, there seems to be no natural systems covering all sporadics. Why not keep the names and remember the history, at least? Perhaps later developments will suggest a good solution.
Finally some comments about notation for other finite groups. Several recommendations in 5.2 really are at variance with general usage. The authors mention Cm for a cyclic group of order m but not Zm! Their term "diagonal product'' A△B is otherwise known as a pullback or a fiber product. The most common notation for an extraspecial group is p1+2n or p1+2nϵ. Since notation for an extension A⋅B reads left-to-right along an ascending series, it would be more appropriate to write (A×B)12 than 12(A×B).
(II): The organization of the individual tables is discussed in Section 6. See page xxiv for a well-diagrammed example. Let G be the simple group. The tables come in blocks with each block corresponding to an extension of the form m.G.a, where m is a cyclic quotient of the Schur multiplier and a is a cyclic subgroup of the outer automorphism group; for reaons why these cases suffice (nearly), see 6.5 and 6.6.
To the left of the block is the downward running list of characters (χ1=1,χ2,χ3,⋯) and their indicators (0, + or − as the character is not real-valued, afforded by a real representation, or real-valued but not afforded by a real representation). Across the top is a band with several rows of information about the columns (indexed by the conjugacy classes, Ci,i=1,⋯,k). The experience of the last 25 years has shown the importance of enriching the traditional "classic'' character table to include power maps (i.e., for n∈Z, which classes contain the nth powers of elements from a fixed class), factorizations (i.e. if g∈Ci and π is a set of primes and g=gπgπ′ is the unique commuting factorization of g into a π-element and a π′-element, which Cj contains gπ), and so on. A simple application of this information, which is not possible to execute with a strictly classical table, is to find the dimension of the space of cubic invariants on a module V affording the character χ. The character on the symmetric tensor cube of V is g↦16{χ(g)3+3χ(g)χ(g2)+2χ(g)3} and so its inner product with the trivial character of G gives the answer.
The difficulty of getting these blocks correct increases generally according to the sequence m=1, a=1; a=1; m,a arbitrary. Indeed the authors acknowledge errors which turned up as the book went to press (see page xxxii, bottom). How the notations extend across the several upward and downward extensions is articulated well.
(III): The final part of the Atlas text consists of three tables and a list of references. (1) Partitions and classes of characters for Sn, useful, say, in working out particular invariants of the group in question. (2) Involvement of sporadic groups in one another (the single "?'' in this Atlas table is now claimed to be "−'' in recent work of R. A. Wilson ). (3) Orders of over 250 simple groups, with orders in base 10 and in factorized forms and with Schur multiplier and outer automorphism group.
(IV) The bibliography is restricted to (i) some very general works on the families of finite simple groups and (ii) lengthy lists of articles on each of the 26 sporadic groups.
Survey articles (no proofs) for absolute beginners are worth mentioning and could go in (i), e.g., a paper by R. Carter [J. London Math. Soc. 40 (1965), 193–240; MR0174655] for groups of Lie type and a paper by the reviewer [in Vertex operators in mathematics and physics (Berkeley, Calif., 1983), 217–229, Springer, New York, 1985; MR0781380] for sporadic groups. Also, references for Schur multiplier and automorphism groups would be of general interest.
Tables of numerical information are notorious for errors and it does pay to compare; for example, the order of McLaughlin's group is incorrectly given on page 136 of D. Gorenstein 's Finite simple groups [Plenum, New York, 1982; MR0698782]. After the Higman-Sims group, G, was discovered in 1968, it was deduced that G must have subgroups K≤H≤G with H≅PSU(3,5) and K≅Alt7. Of course, the characters of G must restrict sensibly to characters of K and H but the character tables then at hand produced a contradiction! The error in the tables was found.
Should a researcher, urgently needing to prove a theorem, trust the Atlas? The question is like that of whether to accept the classification of finite simple groups. Both efforts are widely respected, the participants in both have worked at high levels to reach the goal, yet have admitted that errors exist. In both cases, the group theory community feels that probably only local adjustments would be needed in the ambient program to deal with errors. So, the answer is: "Yes, but…''.
Only a purist would turn his or her back on either claim of completion. To make progress, we must accept them as essentially correct but pay attention for some time and look for alternate arguments whenever possible. One can treat them as axioms when writing arguments down formally.
Norton has shown a list of errors discovered since publication. One is a nonsquare character table! It is worth mentioning that Chat-Yin Ho recently found a maximal 7-local subgroup of the Monster not on the Atlas list. There may be a problem with the list of maximal subgroups for Co1.
{Reviewer's remarks: The reviewer is disappointed at the incorrectness of the scholarship in a few instances (notwithstanding the disclaimer on page xxxii, Section 8.5.1). The correctness of the Monster character table is not completely proved (though not doubted). (a) The determination of the conjugacy classes requires sufficient knowledge of centralizers of elements in a subgroup of M of the form 21+24⋅Co1; the authors guessed the basic information, then proceeded. (b) The existence of the irreducible character of degree 196883 was taken as a hypothesis (196883 is the smallest number which could be the degree of a nonprincipal character); a proof that such a character exists was claimed by Norton in 1981 but no manuscript has appeared, and its relationship with (a) has not been explicitly stated; existence of such a character is necessary to complete the program devised by J. G. Thompson [Bull. London Math. Soc. 11 (1979), no. 3, 340–346; MR0554400] for proving uniqueness of M.
{It would have been helpful to have some recent references, e.g. to the reviewer's recent work on code loops. The reviewer understands that future editions will contain no new references.
{The book is attractive in appearance. The cover is a cherry red with white writing on stiff cardboard. The authors' names form a neat matrix listed vertically in alphabetical order (which agrees with their respective ages, apparently), each with two initials and a 6-letter last name. The price is extremely fair. The authors are to be commended for their influence on the price and for getting the publisher to replace the originally intended soft binding.
{The book is large—too large for most briefcases. The wire binding on the reviewer's copy became deformed right away and interfered with easy closing and opening of the book to lie flat on a table. The edges of the pages near the binding have begun to suffer due to struggles with the binding. One idea is to make the tables available on tape, potentially a big saving of effort for the user who intends computer calculations.
{The mathematics community (and physics community) should be grateful to the creators of the Atlas for their extremely fine service. An appreciation and use of the finite simple groups might be expected to spread noticeably faster as a result.}

Reviewed by R. L. Griess