Lets understand direct product of two subgroups with the help of a problem. This problem is useful for College Mathematics.
Problem: Direct Product of two subgroups
If $ H_1 , H_2 $ are subgroups of a group G then $ H_1 . H_2 = { h_1 h_2 \in G, h_1 \in H_1 , h_2 \in H_2 } $ is a subgroup of G.
False
Discussion: If one of the groups is normal then the above assertion would be true. Suppose $ h_1 , h_1 ' \in H_1 , h_2 , h_2 ' \in H_2 $ then consider the elements $ h_1 h_2 $ and $ h_1 ' h_2 ' $ both of which are members of the set of $ H_1 H_2 $. If $ H_1 H_2 $ is a group then their product will also be a member of $ H_1 H_2 $. That is $ h_1 h_2 h_1 ' h_2 ' \in H_1 H_2 $.
Suppose one of the subgroups, say $ H_1 $ is normal then $ h_2 h_1 ' h_2 ^{-1} \in H_1 $ or there exists $ h_1 '' \in H_1 $ such that $ h_2 h_1 ' = h_1 '' h_2 $. Hence $ h_1 h_2 h_1 ' h_2 ' = h_1 h_1 '' h_2 h_2 ' $ and this element definitely belongs to $ H_1 H_2 $ as $ h_1 h_1 '' \in H_1 h_2 h_2 ' H_2 $. Existence of identity and inverse are easy to prove.
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