182 Charles C. Sims Computational methods for permutation groups 183 TABLE 2 (continued) Degree Permutations 16 P = (1, 16) (2, 3) (4, 5) (6, 7) (8, 9) (10, 11) (12, 13) (14, 15) b = (1, 3) (2, 16) (4, 6) (5, 7) (8, 10) (9, 11) (12, 14) (13, 15) c = (1, 5) (2, 6) (3, 7) (4, 16) (8, 12) (9, 13) (10, 14) (11, 15) d = (1, 9) (2, 10) (3, 11) (4, 12) (5, 13) (6, 14) (7, 15) (8, 16) e = (1, 12, 7, 5) (2, 4, 13, 11) (3, 8, 10, 14) (6, 9) (15) (16) f= (1, 3, 2) (4, 12, 8) (5, 15, 10) (6, 13, 11) (7, 14, 9) (16) g = (1) (2, 3) (4) (5) (6, 7) (8, 12) (9, 13) (10, 15) (11, 14) (16) h = (1, 15) (2, 12) (3) (4, 10) (5) (6) (7, 9) (8) (11) (13) (14) (16) i = (1, 7) (2, 12) (3, 11) (4, 10) (5, 13) (6) (8) (9, 15) (14) (16) j = (1, 14) (2, 13) (3) (4, 11) (5) ( 6) (7, 8) (9) (10) (12) (15) (16) k = (1, 3) (2) (4, 8) (5, 11) (6, 10) (7, 9) (12) (13, 15) (14) (16) 17 a = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17) b = (1) (2, 4, 10, 11, 14, 6, 16, 12, 17, 15, 9, 8, 5, 13, 3, 7) c = (1) (2, 3) (4, 9) (5, 7) (6, 8) (10, 14) (11, 13) (12, 15) (16, 17) 18 19 20 a = (1, 18) (2) (3, 10) (4, 7) (5, 14) (6, 8) (9, 16) (11, 13) (12, 15) (17) a = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19) b = (1) (2, 3, 5, 9, 17, 14, 8, 15, 10, 19, 18, 16, 12, 4, 7, 13, 6, 11) a = (1, 20) (2, 19) (3, 10) (4, 7) (5, 15) (6, 16) (8, 9) (11, 18) (12, 13) (14, 17) is given in the form of two tables. In Table 1 the groups are listed together with some facts about them. In Table 2 we give the generators for these groups which are referred to in Table 1. The groups in Table 1 are listed by degree and for a fixed degree by order. Beyond this the numbering is arbitrary. The order of the group is listed as is the transitivity t, whenever t s 1. If the group is t-fold primitive, this fact is indicated by the letter p following the transitivity. Generating permutations are given for all groups except the alternating and symmetric groups of each degree. The entries in this column refer to permutations in Table 2. For example, a, denotes the first permutation listed under degree 7 in Table 2. No attempt has been made to give generating sets with the fewest possible elements. Permutations in Table 2 should be considered defined on and fixing all integers greater than the degree under which they are listed. A minus sign (-) in the column headed f indicates that the group contains odd permutations. Whenever the group G is doubly primitive so that the subgroup G,, fixing the last integer on which G acts is a primitive group of degree IZ- 1, this subgroup G, is given. The symbol 6G3, for example, refers to the third group of degree 6. In all cases G,, is the actual group listed and not just permutation isomorphic to it. Any primitive group G of degree n -K 60 has a unique minimal normal subgroup N. If n + 20, then N is simple and primitive or elementary abelian. If N is primitive, it is listed, with a dash (-) indicating that N = G. The letters e.a. mean that N is elementary abelian and imprimitive. We note that for groups 1, 2, and 5 of degree 8 N = (aa, b,, cs), for groups 1 to 7 of degree 9 N = (as, bs), and for groups 1 to 20 of degree 16 N = (a16, bls, ~16, dl+. Whenever the group is abstractly isomorphic to a member of one of the families of groups A,, S,, PSL(n, q), PGL (n, q), this fact is noted in the last column. In those cases involving the groups As z PSL(2,4) z PSL(2,5), ,I$, % PGL(2, 5), PSL (2, 7) 2 PSL (3, 2), As z PSL (2, 9) and As z PSL (4, 2), only one of the two or three possible designations is listed. Great care has been taken to ensure the accuracy of these tables. However, the author would appreciate being informed of any errors that may be found. REFERENCES 1. E. R. BENNET: Primitive groups with a determination of the primitive groups of degree 20. Amer. J. Math. 34 (1912), l-20. 2. H. BURCKHARDT and H. VOGT: Sur les groupes discontinues: Groupes de substitutions. Encyclop&die des sciences muthkmatiques pures et uppliquies, Edition Francaise. Algebre, Tome I, Vol. I (Arithmetique), Chapter I, 0 8 (1909). 3. B. FISCHER: Eine Kennzeichnung der symmetrischen Gruppen vom Grade 6 and 7. Math. Z. 95 (1967), 288-298. 4. M. HALL: The Theory of Groups (New York: The Macmillan Company, 1959 ). 5. M. HALL: Automorphisms of Steiner triple systems. Proc. of the Symp. in Pure Math. 6 (1962), 47-66. 6. M. HALL: Block designs. Applied Combinatorial Mathematics 369-405. (New York: John Wiley & Sons, 1964). 7. D. G. HIGMAN: Intersection matrices for finite permutation groups. J. Algebra 6 (1967), 22-42. 8. T. C. HOLYOKE: On the structure of multiply transitive permutation groups. Amer. J. Math. 74 (1952), 787-796. 9. D. R. HUGHES: On t-designs and groups. Amer. J. Math. 87 (1965), 761-778. 10. D. R. HUGHES: Extensions of designs and groups: projective, symplectic and certain affine groups. Math. Z. 89 (1965), 199-205. 11. W. A. MANN<strong>IN</strong>G: Primitive Groups, Part I (Math. and Astron., Vol. I) (Stanford: Stanford Univ. Press, 1921). 12. G. A. MILLER: Collected Works, Vol. I (Urbana: University of Illinois Press, 1935). 13. E. T. PARKER and P. J. NIKOLAI: A search for analogues of the Mathieu groups. Math. Tables Aids Comput. 12 (1958), 3843. 14. E. T. PARKER and K. I. APPEL: On unsolvable groups of degreep = 4q+ 1, p and q primes. Canad. J. Math. 19 (1967), 583-589. 15. C. C. SIMS: Graphs and finite permutation groups. Math. Z. 95 (1967), 76-86. 16. H. WIELANDT: Finite Permutation Groups (New York: Academic Press, 1964). 17. W. J. WONG: Determination of a class of primitive permutation groups. Math. Z. 99 (1967), 235-246. CPA 13
An algorithm related to the restricted Burnside group of prime exponent E. KRAUSE and K. WESTON Introduction. Denote the freest Lie ring of characteristic c on n generators satisfying the mth Engel condition by L(c, n, m). It is a long-standing conjecture, probably introduced by Sanov, that the Restricted Burnside Problem for prime exponentp is equivalent to the problem of nilpotency for L(p, iz, p- 1). In this report we discuss a general algorithm for Lie rings which analogously to the collection process yields L(p, IZ, m) (mop) as the latter yields R(p, n). This algorithm is not only more practical but can be readily used with a computer. For instance we applied the algorithm aided by a Univac 1107 computer for p = 5, n = 2, m = 4, and found L(5, 2,4) [l]. An associated matrix algebra of a Lie algebra. Suppose L is a Lie algebra over a field F spanned by elements /? = {ml, . . . , mk} and Mkxl is the F-space of kX 1 column matrices over F. Define the coordinate mapping fp: L - Mk+r by fs(m) = column of coordinates of m EL with respect to some linear combination of ,!? (i.e. if /3 is not a basis, fs(m) requires a choice between all of the linear combinations of ,8. In this case any fixed choice is sufficient for m). If Tis a linear operator of L let M,(T) designate the matrix whose ith column is fO(T(mi)). Also denote the inner derivation of I E L by g(Q Then the associated F-algebra Ag of L consists of elements from Mkxl under ordinary matrix addition and multiplication is defined by Cl@C, = q9Mf~YC2>>X Cl, Cl, c2E4 (1) (X denotes matrix multiplication). The following theorems are easily verified. THEOREM 1. & is an isomorphism between L and AB if and only if/3 is a basis. THEOREM 2. If L(p,n,m)/Lk(p,n,m) is spanned by B = {ml, . . .,m,) moduIo Lk (p,n,m) then A, is a Lie algebra over GF(p) on n generators satisfying tlze m-th Engel condition if and only if A, 2 L(p, n, m>/Lk(p, n, m>. The associated algebra of a finitely generated Lie algebra. If L is a Lie algebra with generators x1, . . . , x,, spanned by monomials p = {ml, . . . , 13. 185
Change language