Statistics of the solvable subgroups of the symmetric groups Sn with n = 3, 4, 5, 6. Bernd Fiedler Mathematical Institute, University of Leipzig, 1996 ==================================================================== The following tables have been determined by construction of all solvable subgroups of the symmetric groups s3, s4, s5, s6 by means of the well-known algorithm for the generation of the subgroup lattice of a small permutation group which is described in [1]. Each layer of the lattice construction is listed in the tables. We have collected the subgroups of a layer in classes of conjugated subgroups. An entry #class groups/class order/class ---------------------------------------- 2 {6,4} {2,3} means, that there are two classes of conjugated subgroups. The first class consists of 6 subgroups of order 2, the second class consists of 4 subgoups of oder 3. We use the notations #class := number of classes #groups := number of groups. If all solvable subgroups of S(n-1) are known, the possibility arises to obtain a subset of subgroups of Sn by embedding of the subgroups of the S(n-1) into Sn and generating of all conjugated subgroups of these embedded subgroups. Therefore an entry concerning a layer consists of two lines. The first line gives the statistics of such subgroups of the layer which have been developed from embedded subgroups of the S(n-1), the second line gives the statistics of the other subgroups of the layer which do not arise from embedded subgroups. The calculations have been performed by Mathematica using my Mathematica package PERMS. Among other things important algorithms from [1] are implemented in PERMS. A table of all subgroups of s4 can be found for example in [2]. A complete list of all subgroups of s5 is given in [3]. REMARK: The identity group is absent in every of the following tables. ---------------------------------------------------------------------- Group s3: layer #class groups/class order/class #groups ----------------------------------------------------------------- 1 1 {3} {2} 3 1 {1} {3} 1 2 0 {} 0 1 {1} {6} 1 ----------------------------------------------------------------- sum 3 5 Group s4: layer #class groups/class order/class #groups ----------------------------------------------------------------- 1 2 {6,4} {2,3} 10 1 {3} {2} 3 2 1 {4} {6} 4 3 {3,1,3} {4,4,4} 7 3 0 0 2 {1,3} {12,8} 4 4 0 0 1 {1} {24} 1 ----------------------------------------------------------------- sum 10 29 Group s5: layer #class groups/class order/class #groups ----------------------------------------------------------------- 1 3 {10,10,15} {3,2,2} 35 1 {6} {5} 6 2 4 {10,15,5,15} {6,4,4,4} 45 3 {6,10,10} {10,6,6} 26 3 2 {5,15} {12,8} 20 2 {6,10} {20,12} 16 4 1 {5} {24} 5 0 0 ----------------------------------------------------------------- sum 16 153 Group s6: layer #class groups/class order/class #groups ----------------------------------------------------------------- 1 4 {20,45,15,36} {3,2,2,5} 116 2 {20,15} {3,2} 35 2 7 {36,20,60,45,15,45, {10,6,6,4,4,4,6} 281 60} 8 {10,20,60,60,45,45, {9,6,6,6,4,4,4,4} 300 15,45} 3 4 {36,15,45,60} {20,12,8,12} 156 11 {20,10,20,60,15,45, {18,18,18,12,12,8,8, 335 45,45,15,45,15} 8,8,8,8} 4 1 {15} {24} 15 9 {10,10,10,15,15,15, {36,36,36,24,24,24, 150 15,15,45} 24,24,16} 5 0 0 3 {10,15,15} {72,48,48} 40 ----------------------------------------------------------------- sum 49 1428 References ---------- [1] G. Butler. Fundamental Algorithms for Permutation Groups, volume 559 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, Heidelberg, New York, 1991. [2] H. Lugowski and H. J. Weinert. Grundzuege der Algebra, volume I of Mathematisch-Naturwissenschaftliche Bibliothek. B. G. Teubner, Leipzig, 4. edition, 1968. [3] W. Schnee. Ueber vollstaendige Aufzaehlung von Permutationsgruppen. Math. Nachr., 5 : 135 - 138, 1951.