7. The Sylow theorems state the following facts about a finite group G, of order (G| = pkm (with p prime, k a positive integer, and p not dividing m). Syl: There exist subgroups in G of size pk, called Sylow p-subgroups. Sy2: All Sylow p-subgroups in G, for a particular prime p, are conjugate. Sy3: The number of Sylow p-subgroups in G is congruent to 1 modulo p, and this number divides m. a) Consider the symmetric group Sg of permutations of 8 object positions. (i) Give the prime decomposition of the order $8]. (ii) For each prime, p > 3 and dividing the order of Sg, consider Sylow p-subgroups of Sg. Give an explicit example of such a subgroup, using permutations. By counting appropriate permutation cycles, determine how many such Sylow subgroups there are, within Sp. Verify how this fits with the Sylow theorems. (iii) For p= 3 describe the form of a Sylow 3-subgroup of Sg. Give an explicit example using permutations in Sg. (iv) For p = 2 describe the form of a Sylow 2-subgroup of Sg. Give an explicit example using permutations in Sg. You may draw a picture to help present the structure of a Sylow 2-subgroup, in terms of permutations which generate the subgroup. b) Consider the semi-direct product G = (C3 x C3) > D where D = (8 | 88) = Cg for which d acts by conjugation, as the following automorphism of H = C3 C3, in which a and B are generators for the two C3 factors: (that is, a3 = B3 = 1 and a = Ba) ; gg ; & ; gg ;& 82 ;& ?g ;& 282 c82 & . 3 ;& (i) What is the order of G? (ii) Show directly, or otherwise, that da EG has order 8. Deduce that the order of dh is 8, for every non-trivial h EH. (iii) Identify the Sylow 2-subgroups, and Sylow 3-subgroups in G. What is their order? How many of them are there? 7. The Sylow theorems state the following facts about a finite group G, of order (G| = pkm (with p prime, k a positive integer, and p not dividing m). Syl: There exist subgroups in G of size pk, called Sylow p-subgroups. Sy2: All Sylow p-subgroups in G, for a particular prime p, are conjugate. Sy3: The number of Sylow p-subgroups in G is congruent to 1 modulo p, and this number divides m. a) Consider the symmetric group Sg of permutations of 8 object positions. (i) Give the prime decomposition of the order $8]. (ii) For each prime, p > 3 and dividing the order of Sg, consider Sylow p-subgroups of Sg. Give an explicit example of such a subgroup, using permutations. By counting appropriate permutation cycles, determine how many such Sylow subgroups there are, within Sp. Verify how this fits with the Sylow theorems. (iii) For p= 3 describe the form of a Sylow 3-subgroup of Sg. Give an explicit example using permutations in Sg. (iv) For p = 2 describe the form of a Sylow 2-subgroup of Sg. Give an explicit example using permutations in Sg. You may draw a picture to help present the structure of a Sylow 2-subgroup, in terms of permutations which generate the subgroup. b) Consider the semi-direct product G = (C3 x C3) > D where D = (8 | 88) = Cg for which d acts by conjugation, as the following automorphism of H = C3 C3, in which a and B are generators for the two C3 factors: (that is, a3 = B3 = 1 and a = Ba) ; gg ; & ; gg ;& 82 ;& ?g ;& 282 c82 & . 3 ;& (i) What is the order of G? (ii) Show directly, or otherwise, that da EG has order 8. Deduce that the order of dh is 8, for every non-trivial h EH. (iii) Identify the Sylow 2-subgroups, and Sylow 3-subgroups in G. What is their order? How many of them are there
