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Mathematics 667 Homework (due Apr 4) A. Hulpke 42) Let G = A 4 and U = ⟨(1, 2, 3)⟩ ≤ G. Consider the representation δ∶ U → C, (1, 2, 3) a ↦ ζ a where ζ is a primitive 3rd root of unity. Determine (for example as images for generators) the induced representation δ ↑GU . 43) a) Let G be a finite group, N ⊲ G and F = G/N. Let φ be the regular permutation representation of F. Show: G is isomorphic to a subgroup of N ≀φ F. 44∗ ) (GAP, using problem 43) Let N = A 6 and F = C 2 . Determine the isomorphism types of groups G that have a normal subgroup A 6 with a factor group C 2 : 1. Construct (using WreathProduct) the wreath product w ∶= A 6 ≀ C 2 . 2. Let u:=List(ConjugacyClassesSubgroups(w),Representative) be a list of subgroups of w (up to conjugacy). 3. Select (using Filtered) those subgroups from U that have order 720 and a derived subgroup of index 2. 4. You will end up with 5 subgroups. Calculate the number of conjugacy classes (for example using NrConjugacyClasses) for each. This shows that only few of then could be isomorphic. Use IsomorphismGroups amongst those candidates to obtain a list of representative up to isomorphism. You will end up with 4 nonisomorphic groups. Which of them is S 6 ? c) Conclude that Aut(A 6 ) has order at least 1440 and that there exists an outer automorphism (i.e. an automorphism not induced by conjugation) for S 6 . What does this automorphism do? (Note: 6 is the only degree in which S n has an outer automorphism. This is the source of many “sporadic” phenomena.) 45) For a prime p, we denote by C p the cyclic group of order p. a) Show that the iterated wreath product (⋯((C p ≀ C p ) ≀ C p ) ≀ ⋯) ≀ C p ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ m-fold product (i.e. for m = 1, we get C p , for m = 2 we get C p ≀ C p &c.) can be represented by permutations on p m points and show that 2 m−1 its order is p1+p+p +⋯+p . b) Let n ∈ N and k = log p (n). Consider n written as a sum of p-powers k n = ∑ ai pi i=0 Show that the power of p that divides n! is exactly p ⎛k ⎛ i−1 j ⎞⎞ ∑ ai ⋅ ∑ p ⎝ i=1 ⎝ j=0 ⎠⎠ c) (Cauchy) For n ∈ N, describe the structure of the p-Sylow subgroups of S n . Hint: Consider first the case that n is a pure power of p. 46) Show that induction is transitive, i.e. if S ≤ T ≤ G and χ is a character of S then χ ↑GS = (χ ↑TS ) ↑GT Problems marked with ∗ are bonus problems for extra credit.