Group Theory: Sheet 9

\begin{matrix}c \\ ◯ \end{matrix}G. C. Smith 2001

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1. Suppose that α : G → H is a group epimorphism. Suppose that K ≤ G and let \overline{K} denote the group \{(k)α | k ∈ K\}. Suppose that L ≤ H. Let L̂ denote the group {g ∈ G | (g)α ∈ L}. Let N = Ker α.

(a) Suppose that A ≤ G. Show that ˆ \overline{A} = AN.

(b) Suppose that B ≤ H. Show that \overline{B̂} = B.

(c) Suppose that N ≤ C ≤ D ≤ G. Show that |D : C| = |\overline{D} : \overline{C}| and that C ⊴ D if and only if \overline{C} ⊴ \overline{D}.

(d) Suppose that E ≤ F ≤ H. Show that |F : E| = |F̂ : Ê|.

(e) Show that “bar” and “hat” set up an inclusion preserving, index preserving bijection between the subgroups of G which contain N, and the subgroups of H.

2. Suppose that G is a finite group and that N ⊴G. Let p be a prime number and suppose that P ∈ Syl_{p}(N). Show that G = N_{G}(P)N.

3. Suppose that G is a finite group and that p is a prime number. Let P be a Sylow p subgroup of G.

(a) Suppose that N_{G}(P) ≤ H ≤ G. Prove that N_{G}(H) = H.

(b) Suppose that N_{G}(P) ≤ K ≤ G and that N_{G}(P) ≤ L ≤ G. Furthermore suppose that there is g ∈ G such that H^{g} = K. Prove that H = K.