mathematics, posts by tag: commutator - LiveJournal

For any group G, denote its commutator subgroup by [G,G].

1. Let G be a group and H a subgroup which contains [G,G]. Prove that H is a normal subgroup of G.
2. Let G be a nilpotent group of class n ≥ 2. Let x be an element of G and H be the subgroup generated by x and [G,G]. Prove that H is nilpotent of class < n. (Hint: Prove that [H,H] = C²(H) is contained in C³(G).)