Let a group G be the internal direct product of two  its subgroups H\index{1} and H\index{2}. Then H\index{1} and H\index{2} are the  normal subgroups of G and          G/H\index{1}≅H\index{2}   and   G/H\index{2}≅H\index{1}

Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.5: Normal Subgroups
Problem 22E: 22. If and are both normal subgroups of , prove that is a normal subgroup of .
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Let a group G be the internal direct product of two
 its subgroups H\index{1} and H\index{2}. Then H\index{1} and H\index{2} are the
 normal subgroups of G and
         G/H\index{1}≅H\index{2}   and   G/H\index{2}≅H\index{1}.

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