Prove the internal direct product criterion which was mentioned in classed (without proof): Let G be...

 

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Prove the internal direct product criterion which was mentioned in classed (without proof): Let G be a group and G₁,..., Gn be normal subgroups of G. G=G₁ ... Gn if and only if the following two conditions hold: • G=G₁...Gn, where recall G₁...G₁ is the collection of all elements of G of the form gi.... 9n where for each 1≤ i ≤n, gi E Gi. • For all 1 ≤ i ≤n, Gin (G₁...Gi-1G₁+1...Gn) = {id}. Prove the internal direct product criterion which was mentioned in classed (without proof): Let G be a group and G₁,..., Gn be normal subgroups of G. G=G₁ ... Gn if and only if the following two conditions hold: • G=G₁...Gn, where recall G₁...G₁ is the collection of all elements of G of the form gi.... 9n where for each 1≤ i ≤n, gi E Gi. • For all 1 ≤ i ≤n, Gin (G₁...Gi-1G₁+1...Gn) = {id}.