##
**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 |

### Keywords:

finite simple groups; Schur multiplier; sporadic simple groups; alternating groups; classical linear groups; exceptional Chevalley groups; simple groups of order less than \(10^{25}\); character tables; outer automorphism group; covering group; presentation; generators; relations; maximal subgroups; real representation; conjugacy classes### Online Encyclopedia of Integer Sequences:

Orders of non-cyclic simple groups (without repetition).Orders of sporadic simple groups.

Degrees of irreducible representations of Baby Monster group B.

Degrees of irreducible representations of Monster group M.

The 15 supersingular primes: primes dividing order of Monster simple group.

Highest degree of an irreducible representation of symmetric group S_n of degree n.

Orders of Weyl groups of type E_n.

Degrees of irreducible representations of Mathieu group M_11.

Degrees of irreducible representations of Mathieu group M_12.

Degrees of irreducible representations of Mathieu group M_22.

Degrees of irreducible representations of Mathieu group M_23.

Degrees of irreducible representations of Mathieu group M_24.

Degrees of irreducible representations of alternating group A_5.

Degrees of irreducible representations of alternating group A_6.

Degrees of irreducible representations of alternating group A_7.

Degrees of irreducible representations of alternating group A_8.

Degrees of irreducible representations of alternating group A_9.

Degrees of irreducible representations of alternating group A_10.

Degrees of irreducible representations of alternating group A_11.

Degrees of irreducible representations of alternating group A_12.

Degrees of irreducible representations of alternating group A_13.

Degrees of irreducible representations of symmetric group S_5.

Degrees of irreducible representations of symmetric group S_6.

Degrees of irreducible representations of symmetric group S_7.

Degrees of irreducible representations of symmetric group S_8.

Degrees of irreducible representations of symmetric group S_9.

Degrees of irreducible representations of symmetric group S_10.

Degrees of irreducible representations of symmetric group S_11.

Degrees of irreducible representations of symmetric group S_12.

Degrees of irreducible representations of symmetric group S_13.

Degrees of irreducible representations of group L2(7).

Degrees of irreducible representations of group L2(8).

Degrees of irreducible representations of group L2(11).

Degrees of irreducible representations of group L2(13).

Degrees of irreducible representations of group L2(16).

Degrees of irreducible representations of group L2(17).

Degrees of irreducible representations of group L2(19).

Degrees of irreducible representations of group L2(23).

Degrees of irreducible representations of group L2(25).

Degrees of irreducible representations of group L2(27).

Degrees of irreducible representations of group L2(29).

Degrees of irreducible representations of group L2(31).

Degrees of irreducible representations of group L2(32).

Degrees of irreducible representations of group L3(3).

Degrees of irreducible representations of group L3(4).

Degrees of irreducible representations of group L3(5).

Degrees of irreducible representations of group L3(7).

Degrees of irreducible representations of group L3(8).

Degrees of irreducible representations of group L3(9).

Degrees of irreducible representations of group L4(3).

Degrees of irreducible representations of group L5(2).

Degrees of irreducible representations of Suzuki group Suz.

Degrees of irreducible representations of Conway group Co1.

Degrees of irreducible representations of Janko group J1.

Degrees of irreducible representations of Janko group J2.

Degrees of irreducible representations of Janko group J3.

Degrees of irreducible representations of Janko group J4.

Degrees of irreducible representations of Higman-Sims group HS.

Degrees of irreducible representations of McLaughlin group McL.

Degrees of irreducible representations of Conway group Co3.

Degrees of irreducible representations of Conway group Co2.

Degrees of irreducible representations of Held group He.

Degrees of irreducible representations of Fischer group Fi22.

Degrees of irreducible representations of Fischer group Fi23.

Degrees of irreducible representations of Harada-Norton group HN.

Degrees of irreducible representations of Thompson group Th.

Degrees of irreducible representations of Lyons group Ly.

Degrees of irreducible representations of Rudvalis group Ru.

Degrees of irreducible representations of O’Nan group ON.

Orders of simple groups.

Sum of degrees of irreducible representations of alternating group A_n.

Degrees of irreducible representations of group U3(3).

Degrees of irreducible representations of group U3(4).

Degrees of irreducible representations of group U3(5).

Degrees of irreducible representations of group U3(7).

Degrees of irreducible representations of group U3(8).

Degrees of irreducible representations of group U3(9).

Degrees of irreducible representations of group U3(11).

Degrees of irreducible representations of group U4(2).

Degrees of irreducible representations of group U4(3).

Degrees of irreducible representations of group U5(2).

Degrees of irreducible representations of group U6(2).

Orders of non-cyclic simple groups (divided by 4).

List of sizes of conjugacy classes of Mathieu simple group M_24 of order 244823040.

Largest order of even permutation of n elements, or maximal order of element of alternating group A_n.

Irregular triangle read by rows: T(n,k) is the number of elements of alternating group A_n having order k, for n >= 1, 1 <= k <= A051593(n).

Degrees of irreducible representations of symmetric group S_14.

Triangle T(n,k) in which n-th row gives degrees of irreducible representations of symmetric group S_n.

Triangle T(n,k) in which n-th row gives degrees of irreducible representations of alternating group A_n.

Triangle whose rows are the degrees of the irreducible representations of the groups PSL(2,p) as p runs through the primes.

Triangle whose rows are the degrees of the irreducible representations of the groups PSL(2,q) as q runs through the primes and prime powers.

Order of the group GU(n,2), the general unitary n X n matrices over the finite field GF(4).

List of orders of centralizers of conjugacy classes of Mathieu simple group M_24 of order 244823040.

Degrees of irreducible representations of group L3(11).

Products of supersingular primes (A002267).

Degrees of irreducible representations of orthogonal group O7(3).

Degrees of irreducible representations of Suzuki group Sz(8).

Degrees of irreducible representations of Suzuki group Sz(32).

Degrees of irreducible representations of symplectic group S4(4).

Degrees of irreducible representations of symplectic group S4(5).

Degrees of irreducible representations of symplectic group S6(2).

Degrees of irreducible representations of symplectic group S6(3).

Degrees of irreducible representations of symplectic group S8(2).

Degrees of irreducible representations of orthogonal group O8+(2).

Degrees of irreducible representations of orthogonal group O8+(3).

Degrees of irreducible representations of orthogonal group O10+(2).

Degrees of irreducible representations of orthogonal group O8-(2).

Degrees of irreducible representations of orthogonal group O8-(3).

Degrees of irreducible representations of orthogonal group O10-(2).

Degrees of irreducible representations of triality twisted group 3D4(2).

Degrees of irreducible representations of simple Chevalley group E6(2).

Degrees of irreducible representations of twisted simple Chevalley group 2E6(2).

Degrees of irreducible representations of simple Chevalley group E7(2).

Degrees of irreducible representations of simple Chevalley group E8(2).

Degrees of irreducible representations of simple Chevalley group F4(2).

Degrees of irreducible representations of Tits group 2F4(2)’.

Degrees of irreducible representations of simple Chevalley group G2(3).

Degrees of irreducible representations of simple Chevalley group G2(4).

Degrees of irreducible representations of simple Chevalley group G2(5).

Degrees of irreducible representations of Ree group R(27).

Degrees of irreducible representations of Fischer group Fi24’.

Orders of groups in chain of subgroups of Conway’s group Co_0 arising from complete graphs.

The prime divisors of the orders of the sporadic finite simple groups.

The orders, with repetition, of the non-cyclic finite simple groups whose orders are 23-smooth.

The orders, with repetition, of the non-cyclic finite simple groups that are subquotients of the automorphism groups of sublattices of the Leech lattice.

The orders, with repetition, of the non-cyclic finite simple groups that are subquotients of the sporadic finite simple groups.

The intersection of A330584 and A330585.

Order of the finite groups GL(m,q) [or GL_m(q)] in increasing order as q runs through the prime powers.

Orders of centralizers of conjugacy classes of the Monster in Atlas order.

a(n) is the largest base in which the order of the Monster group has (47 - n) zeros; alternatively, radicals of maximal powers dividing the order of the Monster group.

Unique values, or record values, of A343743.