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

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Define a chain of subgroups ϒi,{G) of a group G as follows: 'ϒ1(G) = G, ϒ2{G) = (G,G), ϒi,{G) = {ϒi_1(G),G) (see Exercise 3). Show that G is nilpotent if and only if ϒm{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.

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