Augmentation quotients of semidirect products (Q1057359)

Let G be a semidirect product of the normal subgroup H by a subgroup K. Let \(A_ G\) denote the augmentation ideal of the group ring ZG. Using the N-series \(H=H_{(1)}\supseteq H_{(2)}=[H,G]\supseteq H_{(3)}=[H,G,G]\supseteq...\) of H we prove the theorem: \(A_ KA^ m_ G/A_ KA_ G^{m+1}\simeq A^ m_ K/A_ K^{m+1}\oplus (\Gamma_ m/\Gamma_{m+1}).\) We deduce the group structure of \(A_ KA_ G/A_ KA^ 2_ G\) and \(A_ KA^ 2_ G/A_ KA^ 3_ G\) when G is finite.