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