Define a chain of subgroups i ,{G) of a group G as follows: ' 1 (G)

Define a chain of subgroups ϒ_i,{G) of a group G as follows: 'ϒ₁(G) = G, ϒ₂{G) = (G,G), ϒ_i,{G) = {ϒ_{i_1}(G),G) (see Exercise 3). Show that G is nilpotent if and only if ϒ_m{G) = (e) for some m.

Data from exercise 3

If H and K are subgroups of a group G, let (H,K) be the subgroup of G generated by the elements { hkh^-1k¹| h ϵ H, k ϵ K}. Show that

{a) (H,K) is normal in H V K.

(b) If (H,G') = (e), then (H',G) = (e). (c) H

(c) H ⊲ G if and only if (H,G) < H.

(d) Let K ⊲ G and K < H; then H/K < C(G/K) if and only if (H,G) < K.