Abstract Algebra: Theory and Applications · Problem 15.8.6
Rings — Thomas Judson
Mathematics — Abstract Algebra · sylow-theorems
Problem
The proofs of the Second and Third Sylow Theorems (Theorem 15.1.7, Theorem 15.1.8) employ a group action on sets of Sylow p-subgroups. For the Second Theorem, the list is proposed as incomplete and is proved to be all of the Sylow p-subgroups. In this exercise we will see how these actions behave, and how they are different when we use different groups acting on the same set.
Construct the six Sylow 5-subgroups of the alternating group A5. This will be the set of objects for both of our actions. Conjugating one of these Sylow 5-subgroups by an element of A5 will produce another Sylow 5-subgroup, and so can be used to create a group action. For such an action, from each group element form a Sage permutation of the subgroups by numbering the six subgroups and using these integers as markers for the subgroups. You will find the Python list method .index() very helpful. Now use all of these permutations to generate a permutation group (a subgroup of S6). Finally, use permutation group methods for orbits and stabilisers, etc. to explore the actions.
For the first action, use all of A5 as the group. Show that the resulting action is transitive. In other words, there is exactly one single orbit.
For the second action, use just one of the Sylow 5-subgroups as the group. Write the class equation for this action in a format that suggests the ``congruent to 1 mod p'' part of the conclusion of the Third Theorem.