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.