Sylow theorems are a set of three theorems in group theory that help in finding the number of subgroups in a finite group. These theorems are named after the mathematician Ludwig Sylow.
Groups of order 51 and 39 are important because they are two of the smallest non-cyclic groups. They also have interesting properties and can be used to illustrate the concepts of Sylow theorems.
According to Sylow's first theorem, the number of Sylow 3-subgroups in a group of order 51 must divide 51 and leave a remainder of 1. Therefore, the possible number of Sylow 3-subgroups is 1 or 17. To determine the exact number, we can use Sylow's third theorem which states that the number of Sylow 3-subgroups is congruent to 1 mod 3 and divides 51. Therefore, the only possible number of Sylow 3-subgroups in a group of order 51 is 1.
No, a group of order 39 cannot have a normal Sylow 13-subgroup. This is because the number of Sylow 13-subgroups must divide 39 and leave a remainder of 1 according to Sylow's first theorem. Since 13 does not divide 39, there cannot be a normal Sylow 13-subgroup.
Sylow theorems provide information about the number of subgroups in a group, which in turn can help in determining the structure of the group. For example, if the number of Sylow p-subgroups is 1, then the group is a p-group and has a cyclic subgroup of order p. Sylow theorems also help in finding normal subgroups and the intersection of subgroups, which are important in understanding the structure of a group.