(a) Show that the analogue of Theorem 7.11 is false for nilpotent groups [Consider S₃].

(b) If H < C( G) and G/H is nilpotent, then G is nilpotent.

Theorem 7.11

(i) Every subgroup and every homomorphic image of a solvable group is solvable. (ii) if N is a normal subgroup of a group G such that N and G/N are solvable, then G is solvable.

SKETCH OF PROOF.

{i) If ∫: G →H is a homomorphism [epimorphism], verify that ∫(Gⁱ) < H⁽ⁱ⁾[∫(Gⁱ) = H⁽ⁱ⁾] for all i. Suppose ∫ is an epimorphism, and G is solvable. Then for some n, (e) = ∫(e) = ∫(G⁽ⁿ⁾) = Hⁿ, whence His solvable. The proof for a subgroup is similar.

(ii) Let ∫: G →G/ N be the canonical epimorphism. Since G/ N is solvable, for some n ∫(G⁽ⁿ⁾) = (G/N)⁽ⁿ⁾ = (e). Hence G⁽ⁿ⁾< Ker ∫ = N.Since G⁽ⁿ⁾ is solvable by 

(i), there exists k ϵ N* such that G^(n+k) = (G⁽ⁿ⁾)^(k) = (e). Therefore, G is solvable.