This exercise verifies Theorem 15.13. The commutator subgroup is computed with the permutation group method .commutator(). For the dihedral group of order \(40\text{,}\)\(D_{20}\) (DihedralGroup(20) in Sage), compute the commutator subgroup and form the quotient with the dihedral group. Then verify that this quotient is abelian. Can you identify the quotient group exactly (in other words, up to isomorphism)?
2.
For each possible prime, find all of the distinct Sylow \(p\)-subgroups of the alternating group \(A_5\text{.}\) Confirm that your results are consistent with the Third Sylow Theorem for each prime. We know that \(A_5\) is a simple group. Explain how this would explain or predict some aspects of your answers.
Count the number of distinct elements contained in the union of all the Sylow subgroups you just found. What is interesting about this count?
3.
For the dihedral group \(D_{36}\) (symmetries of a \(36\)-gon) and each possible prime, determine the possibilities for the number of distinct Sylow \(p\)-subgroups as predicted by the Third Sylow Theorem (Theorem 15.8). Now compute the actual number of distinct Sylow \(p\)-subgroups for each prime and comment on the result.
It can be proved that any group with order \(72\) is not a simple group, using techniques such as those used in the later examples in this chapter. Discuss this result in the context of your computations with Sage.
4.
This exercise verifies Lemma 15.6. Let \(G\) be the dihedral group of order \(36\text{,}\)\(D_{18}\text{.}\) Let \(H\) be the one Sylow \(3\)-subgroup. Let \(K\) be the subgroup of order \(6\) generated by the two permutations a and b given below. First, form a list of the distinct conjugates of \(K\) by the elements of \(H\text{,}\) and determine the number of subgroups in this list. Compare this with the index given in the statement of the lemma, employing a single (long) statement making use of the .order(), .normalizer() and .intersection() methods with G, \(H\) and \(K\text{,}\)only.
5.
Example 15.19 shows that every group of order \(48\) has a normal subgroup. The dicyclic groups are an infinite family of non-abelian groups with order \(4n\text{,}\) which includes the quaternions (the case of \(n=2\)). So the permutation group DiCyclicGroup(12) has order 48. Use Sage to follow the logic of the proof in Example 15.19 and construct a normal subgroup in this group. (In other words, do not just ask for a list of the normal subgroups from Sage, but instead trace through the implications in the example to arrive at a normal subgroup, and then check your answer.)
6.
The proofs of the Second and Third Sylow Theorems (Theorem 15.7, Theorem 15.8) 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 \(A_5\text{.}\) This will be the set of objects for both of our actions. Conjugating one of these Sylow \(5\)-subgroups by an element of \(A_5\) 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 \(S_6\)). Finally, use permutation group methods for orbits and stabilisers, etc. to explore the actions.
For the first action, use all of \(A_5\) 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.
