## Finite permutation groups and finite simple groups (1981)

Venue: | Bull. London Math. Soc |

Citations: | 94 - 3 self |

### BibTeX

@ARTICLE{Cameron81finitepermutation,

author = {Peter J. Cameron},

title = {Finite permutation groups and finite simple groups},

journal = {Bull. London Math. Soc},

year = {1981},

volume = {13},

pages = {1--22}

}

### Years of Citing Articles

### OpenURL

### Abstract

In the past two decades, there have been far-reaching developments in the problem of determining all finite non-abelian simple groups—so much so, that many people now believe that the solution to the problem is imminent. And now, as I correct these proofs in October 1980, the solution has just been announced. Of

### Citations

275 |
Every planar map is four-colorable
- Appel, Haken
- 1976
(Show Context)
Citation Context ...of researchers. However, it raises philosophical problems of a very different kind from those posed by a recent well-publicised breakthrough, the proof of the fourcolour conjecture by Appel and Haken =-=[1]-=-: human beings are less reliable than properly-programmed machines. It is certain that the first published proof of (S), spread over many journals'and theses, will contain mistakes. Thus, it is import... |

127 |
Theory of Groups of Finite Order
- Burnside
- 1911
(Show Context)
Citation Context ...ion is related to the well-known fact that the order of any finite group is bounded by a function of the number of its conjugacy classes (the orbits of the group of inner automorphisms). See Burnside =-=[9]-=-, page 461. As a corollary of Proposition 5.1, we obtain the following result of Burnside ([9], page 202); more properly, Proposition 5.1 should be seen as a generalisation of Burnside's theorem.s8 PE... |

83 |
The Graph Isomorphism Disease
- Read, Coreneil
(Show Context)
Citation Context ...reasonable to test the known simple groups for possible irreducible actions—see [30] for some results. Another such instance is the study of computational complexity of graph isomorphism testing (see =-=[52]-=-). Fast algorithms for checking the isomorphism of two graphs often fail if one of the graphs admits non-trivial automorphisms, since such a graph cannot be labelled in a canonical way. So we need to ... |

74 | Solvability of groups of odd order - Feit, Thomson - 1963 |

68 |
Embedding theorems for groups
- Higman, Neumann, et al.
- 1949
(Show Context)
Citation Context ... the proposed subconstituents are representations of the same group (not necessarily faithful) and that the point stabilisers K and L are isomorphic. Form the HNN extension G = (H,t:t~ l Kt = L> (see =-=[33]-=-). Britton's lemma [7] guarantees that H r\t~ x Ht = L, H n tHt' 1 = K, so that G acting on the cosets of H has the desired subconstituents. However, something can be salvaged. If paired constituents ... |

55 |
Hilbert's Tenth Problem. Diophantine Equations: Positive Aspects of a Negative Solution, in: Mathematical developments arising from Hilbert
- Davis, Matiyasevich, et al.
- 1976
(Show Context)
Citation Context ...ns that we could in principle construct a machine which would decide, given a natural number n, whether or not n e M. By the results of Davis, Matijasevic and Robinson on Hilbert's tenth problem (see =-=[18]-=-), we know that M can even be expressed as the set of positive values of a polynomial. However, even if such a polynomial were explicitly known, it would not constitute a satisfactory solution to the ... |

49 |
Sur certains groupes simples
- Chevalley
- 1955
(Show Context)
Citation Context ... of Lie type, or one of finitely many sporadic groups. The alternating groups require little comment: there is one such simple group for each degree n ^ 5. Groups of Lie type are the Chevalley groups =-=[15]-=- and twisted analogues discovered by Steinberg [60], Suzuki [61], Ree [53, 54] and Tits [63]. It will be convenient to divide them into two types: the classical linear, symplectic, unitary and orthogo... |

49 |
Computational methods in the study of permutation groups
- Sims
- 1967
(Show Context)
Citation Context ...ty. Any group element is uniquely determined by its action on a base. For this reason, bases are used extensively in machine computations involving permutation groups (especially in the hands of Sims =-=[59]-=-), and in theoretical investigations of computational complexity. It was in the latter context that Babai showed that a primitive but not 2-transitive group G of degree n has a base of cardinality at ... |

30 |
The 2-transitive permutation representations of the finite Chevalley groups
- Curtis, Kantor, et al.
(Show Context)
Citation Context ...or k ^ 5, k ± 6); all the 2-transitive representations of these groups were determined by Maillet [43]. When N is a group of Lie type, the corresponding problem was solved by Curtis, Kantor and Seitz =-=[17]-=- (see also Howlett [35]). It is implicit in (S) that the sporadic groups can be handled by ad hoc arguments. We illustrate one method of proving theorems of this type, by dealing with the symmetric gr... |

27 |
6-transitive graphs
- Cameron
- 1980
(Show Context)
Citation Context ...at most n vertices extends to an automorphism of F. (This is a specialisation to graphs of diameter 2 of a more general definition due to Meredith [45].) Using purely combinatorial arguments, Cameron =-=[12]-=- determined all 5-tuple transitive graphs. Recently, Buczak [8] has shown that only two further graphs are 4-tuple transitive. His proof uses (S) in a strong sense: detailed knowledge of the simple gr... |

24 |
On the order of uniprimitive permutation groups
- Babai
- 1981
(Show Context)
Citation Context ...c", provided that G is not 2-transitive. (Wielandt gave c = 24.) Praeger and Saxl [50] extended the result to all primitive groups, at the same time reducing Wielandt's constant to 4. Recently, Babai =-=[3]-=- has obtained a substantial improvement: if G is not 2-transitive, then \G\ ^ ,i 4 " l/2|og ". This result is close to best possible, since both the symmetric group Sm acting on unordered pairs (with ... |

23 |
2-transitive and antiflag transitive collineation groups of finite projective spaces
- Cameron, Kantor
- 1979
(Show Context)
Citation Context ...up with a regular normal subgroup. In determining the 3-transitive groups, we may use either the theorem of Hering, Kantor and Seitz [29] (see Theorem 10.2) or the determination by Cameron and Kantor =-=[13]-=- of 3-transitive groups with elementary abelian socle. Both of these theorems are independent of (S);s10 PETER J.CAMERON the group theory used in the proof of the second is elementary, in the sense th... |

23 |
Permutation groups through invariant relations and invariant functions, (Lecture
- Wielandt
(Show Context)
Citation Context ...c expansion of log |G| below that derived from the trivial bound \G\ ^ n\. Using methods similar to Jordan's (that is, bounding the orders of Sylow p-subgroups for all primes p less than n), Wielandt =-=[69]-=- was able to give a substantially better bound: \G\ < c", provided that G is not 2-transitive. (Wielandt gave c = 24.) Praeger and Saxl [50] extended the result to all primitive groups, at the same ti... |

22 |
Finite groups generated by 3-transpositions
- Fischer
- 1971
(Show Context)
Citation Context .... Such hypotheses may be graph-theoretic, involving subdegrees and intersection numbers (as above) or configuration theorems (see Shult [57]), or group-theoretic, as in the following example. Fischer =-=[22]-=- has determined the finite groups generated by a conjugacy class of 3-transpositions (involutions with the property that the product of any two of them has order 1, 2 or 3). Under suitable hypotheses ... |

21 | On trivalent graphs
- Biggs, Smith
- 1971
(Show Context)
Citation Context ... acts transitively on pairs of vertices at any given distance. It is conjectured that there are only finitely many such graphs with any given valency. This was proved for valency 3 by Biggs and Smith =-=[6]-=-, and the truth of Sims' conjecture was crucial for the proof. Another example is the use of the Feit-Thompson theorem in the determination of distance-transitive directed graphs of odd girth [5]. A g... |

17 |
Finite permutation groups of rank 3
- Higman
- 1964
(Show Context)
Citation Context ...re algebraic functions of the intersection numbers which must be nonnegative integers. This so-called integrality condition has been important in the study of primitive permutation groups (see Higman =-=[31]-=-, for example). Some information about the intersection numbers can be obtained from knowledge of the degrees and multiplicities: see the paper of Wielandt [66] on groups of degree 2p cited earlier.s1... |

16 |
Seitz - Finite groups with a split BN-pair of rank 1
- Hering, Kantor, et al.
- 1972
(Show Context)
Citation Context ...y (see Wielandt [68], page 28) that S4 is the only 4-transitive group with a regular normal subgroup. In determining the 3-transitive groups, we may use either the theorem of Hering, Kantor and Seitz =-=[29]-=- (see Theorem 10.2) or the determination by Cameron and Kantor [13] of 3-transitive groups with elementary abelian socle. Both of these theorems are independent of (S);s10 PETER J.CAMERON the group th... |

16 |
A family of simple groups associated with the simple Lie algebra of Type
- Ree
- 1961
(Show Context)
Citation Context ...ups require little comment: there is one such simple group for each degree n ^ 5. Groups of Lie type are the Chevalley groups [15] and twisted analogues discovered by Steinberg [60], Suzuki [61], Ree =-=[53, 54]-=- and Tits [63]. It will be convenient to divide them into two types: the classical linear, symplectic, unitary and orthogonal groups (which fall into six families, each parametrised by a dimension n a... |

15 | The rank 3 permutation representations of the finite classical groups
- KANTOR, LIEBLER
(Show Context)
Citation Context ...ry used in the proof of the second is elementary, in the sense that it uses nothing more than Sylow's theorem. We note two further results of the same type as Theorem 5.3. THEOREM 5.5 (Kantor-Liebler =-=[41]-=-). All primitive permutation groups of rank 3 whose socle is a classical simple group are known. A similar result for groups of rank at most 5 whose socle is an alternating group was given by Bannai [... |

11 |
Permutation representations of the finite classical groups of small degree or rank
- Kantor
- 1979
(Show Context)
Citation Context ...n a positive integer r, all but finitely many primitive permutation groups of rank r with simple socle are known. This was proved by Bannai [4] when the socle N is an alternating group, and by Kantor =-=[40]-=- (depending on results of Seitz [55]) in case N is of Lie type; the sporadic groups are irrelevant here. The idea of the proof is that, as above, if G is primitive and has a conjugacy class C, then Ga... |

11 | On the orders of primitive permutation groups
- Praeger, Saxl
- 1980
(Show Context)
Citation Context ...f Sylow p-subgroups for all primes p less than n), Wielandt [69] was able to give a substantially better bound: \G\ < c", provided that G is not 2-transitive. (Wielandt gave c = 24.) Praeger and Saxl =-=[50]-=- extended the result to all primitive groups, at the same time reducing Wielandt's constant to 4. Recently, Babai [3] has obtained a substantial improvement: if G is not 2-transitive, then \G\ ^ ,i 4 ... |

11 |
Graphs and finite permutation groups
- Sims
- 1968
(Show Context)
Citation Context ... This enables us to label the suborbits in a consistent way so that, for example, T{ag) = T{a)g for all geG. Now the pairing of orbitals induces a pairing of suborbits. It was suggested by C. C. Sims =-=[58]-=- that we should consider an orbital F as the set of edges of a directed graph with vertex set Cl. Clearly G is a group of automorphisms of this graph, acting transitively on vertices and on edges. If ... |

10 |
groupes simples de Suzuki et de Ree
- Les
- 1960
(Show Context)
Citation Context ... comment: there is one such simple group for each degree n ^ 5. Groups of Lie type are the Chevalley groups [15] and twisted analogues discovered by Steinberg [60], Suzuki [61], Ree [53, 54] and Tits =-=[63]-=-. It will be convenient to divide them into two types: the classical linear, symplectic, unitary and orthogonal groups (which fall into six families, each parametrised by a dimension n and a field ord... |

9 |
The orders of the classical simple groups
- Artin
- 1955
(Show Context)
Citation Context ...wn simple groups are characterised (within the class of all known simple groups) by their orders, with known exceptions: |PSp(2m,g)| = |PQ(2m + \,q)\ for m ^ 3 and q odd, and |PSL(3,4)| = \A8\ (Artin =-=[2]-=-). Does a similar statement hold for characteristically simple groups (direct products of isomorphic simple groups)? Artin's technique should enable this problem to be settled. For a classical group T... |

8 |
Maximal subgroups of low rank of finite symmetric and alternating groups
- Bannai
(Show Context)
Citation Context ...]). All primitive permutation groups of rank 3 whose socle is a classical simple group are known. A similar result for groups of rank at most 5 whose socle is an alternating group was given by Bannai =-=[4]-=-. THEOREM 5.6 (S). Given a positive integer r, all but finitely many primitive permutation groups of rank r with simple socle are known. This was proved by Bannai [4] when the socle N is an alternatin... |

8 |
Zweifach transitive, auflösbare Permutationsgruppen
- Huppert
- 1957
(Show Context)
Citation Context ...l subgroup N of such a group G is elementary abelian. Then Ga is a subgroup of GL (d, p), transitive on nonzero vectors of the vector space. The soluble 2-transitive groups were determined by Huppert =-=[36]-=-, so we may assume that Ga is insoluble. Hering [26] showed that Ga has a unique nonabelian composition factor, and subsequently [28] he examined the known simple groups and determined all such situat... |

8 |
rank permutation representations of finite Chevalley groups
- Seitz, Small
- 1974
(Show Context)
Citation Context ...tely many primitive permutation groups of rank r with simple socle are known. This was proved by Bannai [4] when the socle N is an alternating group, and by Kantor [40] (depending on results of Seitz =-=[55]-=-) in case N is of Lie type; the sporadic groups are irrelevant here. The idea of the proof is that, as above, if G is primitive and has a conjugacy class C, then Ga has an orbit of size at most \C\, w... |

7 |
The word problem
- Britton
- 1963
(Show Context)
Citation Context ...tuents are representations of the same group (not necessarily faithful) and that the point stabilisers K and L are isomorphic. Form the HNN extension G = (H,t:t~ l Kt = L> (see [33]). Britton's lemma =-=[7]-=- guarantees that H r\t~ x Ht = L, H n tHt' 1 = K, so that G acting on the cosets of H has the desired subconstituents. However, something can be salvaged. If paired constituents in a transitive group ... |

6 |
Variations on a theme of Chevalley
- Steinberg
- 1959
(Show Context)
Citation Context ...ps. The alternating groups require little comment: there is one such simple group for each degree n ^ 5. Groups of Lie type are the Chevalley groups [15] and twisted analogues discovered by Steinberg =-=[60]-=-, Suzuki [61], Ree [53, 54] and Tits [63]. It will be convenient to divide them into two types: the classical linear, symplectic, unitary and orthogonal groups (which fall into six families, each para... |

5 | Permutation groups with multiply transitive suborbits
- CAMERON
- 1972
(Show Context)
Citation Context ...hese need not be isomorphic, even as abstract groups; but there are some relations between them. For example, Sims (see [51]) showed that they have a common non-trivial homomorphic image, and Cameron =-=[10]-=- that if one is 2-transitive then so is the other. Stronger relationships hold in primitive groups; exactly how strong, is not known. We conclude this section with a problem to which (S) is relevant. ... |

5 |
The uniqueness and non-uniqueness of triangular association schemes, Science Record 3
- Chang
- 1959
(Show Context)
Citation Context ...roup G2(2). The chief difficulty in the proof of the theorem (and the reason for the exceptions) is the necessity to show that \i = 4 in order to apply a graph-theoretic characterisation due to Chang =-=[16]-=- and Hoffman [34]. (Here \i is the intersection number |F(a) n T(j5)|, where F(a) is the Ga-orbit of size 2(n-2), and /? £ T(a) u {a}.) But, if the group G is given, it is a trivial computational matt... |

5 |
Characterization of families of rank 3 permutation groups by the subdegree
- Higman
- 1970
(Show Context)
Citation Context ...eat computational burden. These points will be illustrated by examples. The simplest tests to apply to rank 3 groups are characterisations by subdegrees. An example is the following theorem of Higman =-=[32]-=-. THEOREM 10.1. Let G be a permutation group of degree n = \m(m — 1), in which Ga has orbit lengths 1, 2(m —2), and j(m — 2)(m — 3). If m is not one of a known finite list of exceptional values, then ... |

5 |
Characterizations of certain classes of graphs
- Shult
- 1972
(Show Context)
Citation Context ...k 3 groups under hypotheses which are easily checked by computer. Such hypotheses may be graph-theoretic, involving subdegrees and intersection numbers (as above) or configuration theorems (see Shult =-=[57]-=-), or group-theoretic, as in the following example. Fischer [22] has determined the finite groups generated by a conjugacy class of 3-transpositions (involutions with the property that the product of ... |

4 |
Nonexistence of certain distance-transitive digraphs
- Bannai, Cameron, et al.
- 1981
(Show Context)
Citation Context ...mith [6], and the truth of Sims' conjecture was crucial for the proof. Another example is the use of the Feit-Thompson theorem in the determination of distance-transitive directed graphs of odd girth =-=[5]-=-. A graph F is n-tuple transitive if any isomorphism between induced subgraphs on at most n vertices extends to an automorphism of F. (This is a specialisation to graphs of diameter 2 of a more genera... |

4 |
On linear groups which contain an irreducible subgroup of prime order
- HERING
- 1973
(Show Context)
Citation Context .... Then Ga is a subgroup of GL (d, p), transitive on nonzero vectors of the vector space. The soluble 2-transitive groups were determined by Huppert [36], so we may assume that Ga is insoluble. Hering =-=[26]-=- showed that Ga has a unique nonabelian composition factor, and subsequently [28] he examined the known simple groups and determined all such situations in which each could occur as such a composition... |

4 |
On the structure of finite collineation groups of projective planes
- Hering
- 1979
(Show Context)
Citation Context ...ed knowledge of the simple groups and their properties is required. If transitivity is not assumed, it becomes more difficult to apply (S), but this can be done in some instances. For example, Hering =-=[27]-=- showed that, if a group G acts irreducibly on a finite projective plane (fixing no point, line, triangle, or proper subplane), and G contains perspectivities, then the socle of G is either elementary... |

4 |
On the degrees of Steinberg characters of Chevalley groups
- HOWLETT
- 1974
(Show Context)
Citation Context ...he 2-transitive representations of these groups were determined by Maillet [43]. When N is a group of Lie type, the corresponding problem was solved by Curtis, Kantor and Seitz [17] (see also Howlett =-=[35]-=-). It is implicit in (S) that the sporadic groups can be handled by ad hoc arguments. We illustrate one method of proving theorems of this type, by dealing with the symmetric groups. Suppose G = Sk, a... |

4 |
Sur les isomorphes holoédriques et transitifs des groupes symétriques ou alternés
- Maillet
(Show Context)
Citation Context ...er case (when N is simple), we have G ^ Aut (N). If N is an alternating group Ak, then G = Ak or Sk (for k ^ 5, k ± 6); all the 2-transitive representations of these groups were determined by Maillet =-=[43]-=-. When N is a group of Lie type, the corresponding problem was solved by Curtis, Kantor and Seitz [17] (see also Howlett [35]). It is implicit in (S) that the sporadic groups can be handled by ad hoc ... |

4 | Primitive permutation groups with doubly transitive subconstituent - Praeger - 1988 |

4 |
A new type of simple groups of finite order
- Suzuki
- 1960
(Show Context)
Citation Context ...nating groups require little comment: there is one such simple group for each degree n ^ 5. Groups of Lie type are the Chevalley groups [15] and twisted analogues discovered by Steinberg [60], Suzuki =-=[61]-=-, Ree [53, 54] and Tits [63]. It will be convenient to divide them into two types: the classical linear, symplectic, unitary and orthogonal groups (which fall into six families, each parametrised by a... |

4 |
Bounds for the orders of maximal subgroups
- Thompson
- 1970
(Show Context)
Citation Context ...n of k. Certainly the transitive constituent of Ga of degree k has order at most k\\ the conjecture asserts that the kernel K(<x) of the action of Ga on this suborbit F(a) has bounded order. Thompson =-=[62]-=- took an important step towards proving the conjecture when he showed that, under these hypotheses, Ga has a normal subgroup P of prime-power order whose index is bounded by a function g(k) of the sub... |

4 |
As it might have been
- Todd
- 1970
(Show Context)
Citation Context ...lying vector space. (In the exceptional cases, there are additional automorphisms, induced by dualities of the projective space for PSL(d, q) and triality maps of the quadric for PQ + (8,g); see Todd =-=[64]-=- for a geometric description of the additional automorphisms of PSp(4, q) for q even.) If H is reducible, it is the stabiliser of a subspace; if it is irreducible, bounds due to Kantor [40] show that ... |

4 |
Finite permutation groups, Acad
- WIELANDT
- 1964
(Show Context)
Citation Context ... G, giving the right regular representation of G. It can be shown that a permutation group is transitive if and only if its centraliser in the symmetric group is semiregular, and vice versa (Wielandt =-=[68]-=-, page 9). In particular, the centraliser of a regular group is regular. Specifically, the centraliser of the right regular representation is the left regular representation, in which a group element ... |

3 |
Homogeneity conditions in graphs
- Gardiner
- 1978
(Show Context)
Citation Context ...ight expect (S) to be more useful if additional transitivity is required. Three ways in which this has been done are s-arc transitivity, distance-transitivity, and n-tuple transitivity. (See Gardiner =-=[24]-=- for a survey.) An s-arc in a graph is a sequence (a0,..., as) of vertices in which a, and <xi+l are adjacent and a, =/= ai + 2 for all relevant i. In an s-arc transitive undirected graph (one whose a... |

3 |
Perspectivities in irreducible collineation groups of projective planes
- Hering, Walker
- 1977
(Show Context)
Citation Context ...G contains perspectivities, then the socle of G is either elementary abelian of order 9 or non-abelian simple. So it is reasonable to test the known simple groups for possible irreducible actions—see =-=[30]-=- for some results. Another such instance is the study of computational complexity of graph isomorphism testing (see [52]). Fast algorithms for checking the isomorphism of two graphs often fail if one ... |

3 |
On the uniqueness of the triangular association scheme
- Hoffman
- 1960
(Show Context)
Citation Context ...hief difficulty in the proof of the theorem (and the reason for the exceptions) is the necessity to show that \i = 4 in order to apply a graph-theoretic characterisation due to Chang [16] and Hoffman =-=[34]-=-. (Here \i is the intersection number |F(a) n T(j5)|, where F(a) is the Ga-orbit of size 2(n-2), and /? £ T(a) u {a}.) But, if the group G is given, it is a trivial computational matter to check wheth... |

3 |
A theorem concerning simply transitive primitive groups
- Manning
- 1929
(Show Context)
Citation Context ...one is 2-transitive then so is the other. Stronger relationships hold in primitive groups; exactly how strong, is not known. We conclude this section with a problem to which (S) is relevant. Manning (=-=[44]-=-, see also [10], [51]) showed that, if G is primitive but not 2-transitive, and if Ga acts 2-transitively on a suborbit F(a) with subdegree /c, then there is a suborbit A(a) with subdegree / = k{k-l)/... |

2 |
Oeuvres mathematiques: Lettre a M. Auguste Chevalier
- Galois
(Show Context)
Citation Context ...up of Lie type of rank greater than 1, then T contains a subgroup PSL(2, q) or SL(2, q). For q ± 9, a non-trivial permutation representation of this subgroup has degree at least q (a result of Galois =-=[23]-=- for q prime and Dickson [19] in general); so n ^ q. Also \G\ ^ q k for a fixed constant k; so \G\ < n k . The case q = 9 can be treated similarly, or the finitely many exceptional groups over GF(9) r... |

2 |
private communication
- Knapp
(Show Context)
Citation Context ...egularly on the suborbit T(a) f {a}, then Ga acts faithfully on T(a) (that is, |r(a)| = |GJ) (Wielandt [68], page 50). The conjecture is true under additional hypotheses, such as the solubility of Ga =-=[42]-=-. The subrank of a transitive group is defined to be the maximum rank of the transitive constituents of the stabiliser of a point. The author has conjectured that, if G is a primitive group with subra... |

2 |
On multiply transitive groups
- Nagao
- 1965
(Show Context)
Citation Context ...present it seems very likely that Schreier's conjecture will be proved in this way rather than directly.) Wielandt [67] reduced the first of these problems to the second: his result, refined by Nagao =-=[46]-=- and O'Nan [48], asserts that a 6-transitive group must be symmetric or alternating, provided that the composition factors of its proper subgroups have soluble outer automorphism groups. However, unle... |