Solutions for Chapter 4.5 Problem 28E
We know that, a -group is a group of order for. Subgroups of which are -groups, are called -subgroups. If is a group of order, , then a subgroup of order is called a Sylow -subgroup of .
We have been given that, is a group with. Now by the divisibility and congruence criteria of Sylow’s theorem an element is a set of and is a set of . is a unique Sylow 3-subgroup of .
We know that, , where is group of order or . Therefore, we have , and hence and.
Let be a group of order . Now by Sylow’s theorem, has a unique Sylow 7-subgroup . An element preimages under the natural projection.
Thus, and elements map to .