Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With comput. assist. from J. G. Thackray.

*(English)*Zbl 0568.20001
Oxford: Clarendon Press. XXXIII, 252 p. £35.00 (1985).

This book (which in the remainder of this report we will just call the ATLAS) is in its main part a table collection on finite simple groups. Its purpose is to convey detailed information about specific groups. The following informations about a simple group G are provided:

(1) The order of G, the structure of its Schur multiplier, and its outer automorphism group.

(2) Maximal subgroups of G respectively X (G\(\leq X\leq Aut(G)).\)

(3) Various constructions of G respectively of groups Y, where \(Z(Y)\leq Y^{(\infty)}\), and Y/Z(Y) is isomorphic to a group X (G\(\leq X\leq Aut(G)).\)

(4) Presentations of G or Y (Y as in (3)).

(5) Character tables of all groups Y (Y as in (3)).

The only mayor information the ATLAS does not provide are modular characters. The list of finite simple groups in the ATLAS has about 100 members and includes all 26 sporadic simple groups. The alternating groups occur up to degree 13. From the classical linear groups the ”first members” of a family are listed; this means that the Lie-rank is less or equal to 5 and the field is small (i.e. if the Lie-rank is \(\geq 3\), then the field is GF(2) or GF(3)). The exceptional Chevalley groups are only included for small fields (i.e. GF(2) if the Lie-rank is \(\geq 3)\) too. Thus with the exception of small rank Chevalley groups all simple groups of order less than \(10^{25}\) are listed.

There are obvious difficulties to present such an impressive amount of material economically. The authors of the ATLAS were forced to find a compromise between the necessity of presenting the material completely while keeping the size of the book to a minimum, and to make the tables understandable for a wider readership. Thus in the introduction to the ATLAS there is first a small general description of the finite simple groups and the way they were originally constructed. In the second part of the introduction the authors introduce some terminology of their own to give the desired economic presentation. The critical point in understanding the tables in the main part of the ATLAS lies in the readability of this terminology. In particular the notation for character tables of groups Y, where Y has both nontrivial Schur multiplier and outer automorphism group, becomes quite intricate. To understand these tables fully will not be always immediate! However there are some character tables in this introduction, which are presented in the usual way and secondly in the new ”stenograph type” notation.

In the main part of the ATLAS for each of the listed groups there is an extra section. First a group G is listed by all different names under which it occurs thus making exceptional isomorphisms visible (for instance \(A_ 6\) is also listed as \(L_ 2(9)\), \(U_ 2(9)\), \(S_ 2(9)\), \(O_ 3(9)\), \(O^-_ 4(3)\), \(S_ 4(2)'\), \(O_ 5(2)')\). Secondly methods of constructing G (or a covering group) are given. This is done in a very brief way. The reader must usually put in some extra work to fill in the details. Then a presentation with generators and relations is given (this presentation is one which is easy to compute with but normally not one with a minimum number of generators or relations). The maximal subgroups (normally all) are given not only in a structural description but also in the role they play in various constructions. The largest part for a group G will usually be the character tables for the groups Y with Y/Z(Y)\(\simeq X\) (where \(G\leq X\leq Aut(G))\). These tables contain even subtle details: the Schur indicator (which tells us whether a character is nonreal, or real but comes from a nonreal representation, or a character of a real representation) or the conjugacy classes in which the powers of a given representative of a conjugacy class fall etc.

The ATLAS closes with some appendices among which one finds a comprehensive bibliography on finite, sporadic simple groups.

One should emphasize that the ATLAS gives information about individual groups. Usually it should be difficult to extrapolate from the numerical data the ATLAS provides to a whole infinite series of finite simple groups. The ATLAS gives a kind of information, which in published papers on finite simple groups is hard to get, but which is so useful in a concrete situation. The ATLAS should provide for a large readership of mathematicians the quickest and most efficient way to get that kind of information. It also might be just fun (like for the reviewer) to run at random through the ATLAS and find new surprising details about groups which seemed familiar. In conclusion: This ATLAS will be of greatest importance for any mathematician who faces concrete problems in finite simple groups.

(1) The order of G, the structure of its Schur multiplier, and its outer automorphism group.

(2) Maximal subgroups of G respectively X (G\(\leq X\leq Aut(G)).\)

(3) Various constructions of G respectively of groups Y, where \(Z(Y)\leq Y^{(\infty)}\), and Y/Z(Y) is isomorphic to a group X (G\(\leq X\leq Aut(G)).\)

(4) Presentations of G or Y (Y as in (3)).

(5) Character tables of all groups Y (Y as in (3)).

The only mayor information the ATLAS does not provide are modular characters. The list of finite simple groups in the ATLAS has about 100 members and includes all 26 sporadic simple groups. The alternating groups occur up to degree 13. From the classical linear groups the ”first members” of a family are listed; this means that the Lie-rank is less or equal to 5 and the field is small (i.e. if the Lie-rank is \(\geq 3\), then the field is GF(2) or GF(3)). The exceptional Chevalley groups are only included for small fields (i.e. GF(2) if the Lie-rank is \(\geq 3)\) too. Thus with the exception of small rank Chevalley groups all simple groups of order less than \(10^{25}\) are listed.

There are obvious difficulties to present such an impressive amount of material economically. The authors of the ATLAS were forced to find a compromise between the necessity of presenting the material completely while keeping the size of the book to a minimum, and to make the tables understandable for a wider readership. Thus in the introduction to the ATLAS there is first a small general description of the finite simple groups and the way they were originally constructed. In the second part of the introduction the authors introduce some terminology of their own to give the desired economic presentation. The critical point in understanding the tables in the main part of the ATLAS lies in the readability of this terminology. In particular the notation for character tables of groups Y, where Y has both nontrivial Schur multiplier and outer automorphism group, becomes quite intricate. To understand these tables fully will not be always immediate! However there are some character tables in this introduction, which are presented in the usual way and secondly in the new ”stenograph type” notation.

In the main part of the ATLAS for each of the listed groups there is an extra section. First a group G is listed by all different names under which it occurs thus making exceptional isomorphisms visible (for instance \(A_ 6\) is also listed as \(L_ 2(9)\), \(U_ 2(9)\), \(S_ 2(9)\), \(O_ 3(9)\), \(O^-_ 4(3)\), \(S_ 4(2)'\), \(O_ 5(2)')\). Secondly methods of constructing G (or a covering group) are given. This is done in a very brief way. The reader must usually put in some extra work to fill in the details. Then a presentation with generators and relations is given (this presentation is one which is easy to compute with but normally not one with a minimum number of generators or relations). The maximal subgroups (normally all) are given not only in a structural description but also in the role they play in various constructions. The largest part for a group G will usually be the character tables for the groups Y with Y/Z(Y)\(\simeq X\) (where \(G\leq X\leq Aut(G))\). These tables contain even subtle details: the Schur indicator (which tells us whether a character is nonreal, or real but comes from a nonreal representation, or a character of a real representation) or the conjugacy classes in which the powers of a given representative of a conjugacy class fall etc.

The ATLAS closes with some appendices among which one finds a comprehensive bibliography on finite, sporadic simple groups.

One should emphasize that the ATLAS gives information about individual groups. Usually it should be difficult to extrapolate from the numerical data the ATLAS provides to a whole infinite series of finite simple groups. The ATLAS gives a kind of information, which in published papers on finite simple groups is hard to get, but which is so useful in a concrete situation. The ATLAS should provide for a large readership of mathematicians the quickest and most efficient way to get that kind of information. It also might be just fun (like for the reviewer) to run at random through the ATLAS and find new surprising details about groups which seemed familiar. In conclusion: This ATLAS will be of greatest importance for any mathematician who faces concrete problems in finite simple groups.

Reviewer: U.Dempwolff

##### MSC:

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

20-04 | Software, source code, etc. for problems pertaining to group theory |

20D08 | Simple groups: sporadic groups |

20Cxx | Representation theory of groups |

20D05 | Finite simple groups and their classification |

20D06 | Simple groups: alternating groups and groups of Lie type |

20D30 | Series and lattices of subgroups |

20G40 | Linear algebraic groups over finite fields |