Math 551, Assignment 2, due Friday, September 27 in class

1. Let G act on Ω, ω ∈ Ω and H ≤ G. Show that H acts transitively on Ω if and only if G acts transitively on Ω and G = G_{ω}H.

2. (The “Frattini Argument”) Let N be a finite normal subgroup of G, and let P ∈ Syl_{p}(N) for some prime p. Show that G = N_{G}(P)N.

3. Let G be a finite group, G ≠ 1. Define Φ(G) (the “Frattini subgroup” of G) to be the intersection of all the maximal subgroups of G. Show that Φ(G) ◁ G and that every Sylow p-subgroup of Φ(G) for every prime p is also normal in G.

4. Let H ≤ G and set K = ⋂_{g∈G} ^{g}H. Show that K ◁ G and that any normal subgroup of G lying in H actually lies in K.

5. A finite group G is called nilpotent if and only if for each subgroup H ≤ G such that H ≠ G, it is the case that H ≠ N_{G}(H). Show that any group of prime power order p^{a} is nilpotent. (Hint. Consider the conjugation action of H on the set Ω of all G-conjugates of H.)

6. Let σ = [ \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ], τ = [ \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ] , and Q_{8} = ⟨σ, τ⟩ ≤ SL_{2}(C), the

“quaternion” group. Show that

a) Q is a non-abelian group of order 8 in which every subgroup is normal

b) If H is any group such that H = ⟨s, t⟩, s^{4} = 1, t^{2} = s^{2} and tst^{−1} = s^{−1}, then there is a surjective homomorphism Q → H

c) There are exactly two isomorphism classes of nonabelian groups of order 8, namely those represented by D_{8} and Q. (Hint. By a previous exercise, if |G| = 8 and K ≤ G with |K| = 4, then K ◁ G.)

7. Show that any group of order 56 has a normal Sylow 2-subgroup, and that Σ_{7} has no subgroup of order 56. (Hint for second part: If H ≤ Σ_{7} with |H| = 56, then in the action of H on 7 letters, what is the stabilizer of a point?)

8. Let G = GL_{3}(Z/2Z). Show that |G| = 168, and that the upper triangular subgroup T is a Sylow 2-subgroup of G. Show that every element of G of order 2 is conjugate to

t = [ \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ] .

Show that C_{G}(t) = T, and that G has 21 elements of order 2, 42 elements of order 4, 48 elements of order 7, and (for extra credit) 56 elements of order 3.

9. Let G be a finite group and let p be a prime divisor of |G|. Let S(p) = {H | H ≤ G and |H| = p}. Show that |S(p)| ≡ 1 mod p. (Hint. Let a Sylow p-subgroup act by conjugation.)

10. Let n ≥ 4 and G = Σ_{n}. Begin by taking an arbitrary element v ∈ G of order 2 and computing |C_{G}(v)| in terms of the number of disjoint 2-cycles in the cycle decomposition of v.

a) Show that if t = (ab) ∈ G is a transposition and u ∈ G is any element of order 2 such that |C_{G}(t)| = |C_{G}(u)|, then u is also a transposition, unless n = 6.

b) Show that if α ∈ Aut(G) and t ∈ G is a transposition, then α(t) is a transposition.

c) Show that if t = (ab) and u = (cd) are transpositions, then {a, b} ∩ {c, d} consists of a single point if and only if |tu| = 3.

d) Using b) and c), show that if n ≠ 6, then Aut(G) = Inn(G) ≅ G.

e) Show that if n = 6, then Aut(G) ≠ Inn(G), by doing the appropriate exercise in Lang, at the end of chapter 1.