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By Frédérique Oggier

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11 Solvable and nilpotent groups Let us start by introducing a notion stronger than normality. 32. A subgroup H of the group G is called characteristic in G if for each automorphism f of G, we have f (H) = H. We may write H char G. This is stronger than normal since normality corresponds to choose for f the conjugation by an element of g. Note that f restricted to H a characteristic subgroup (denoted by f |H ) is an automorphism of H (it is an endomorphism by definition of H being characteristic).

On the other hand, if P is a Sylow p-subgroup, then it also contains p2 elements, and all of them have order not q, so that we can conclude that P actually contains all elements of order not q, which implies that we have only one Sylow p-subgroup, yielding the wanted contradiction. 50 CHAPTER 1. 3: Cp refers to a cyclic group of prime order. • nq = p : We know from Sylow Theorems that nq ≡ 1 mod q ⇒ p ≡ 1 mod q ⇒ p > q, but also that np | q and since q is prime, that leaves np = 1 or np = q and thus np = q.

Consider the homomorphism a → (aN )(H/N ) which is the composition of the canonical projection π of G onto G/N , and the canonical projection of G/N onto (G/N )/(H/N ) (the latter makes sense since H/N G/N ). We now want to show that H is the kernel of this map, which will conclude the proof since the kernel of a group homomorphism is normal. An element a is in the kernel if and only if (aN )(H/N ) = H/N , that is if and only if aN ∈ H/N , or equivalently aN = hN for some h ∈ H. Since N is contained in H, this means aN is in H and thus so is a, which is what we wanted to prove.

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