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Prove that the intersection of two subgroups is a subgroup. Let A, B are two subgroup of group G, prove that AB is subgroup of G. Proof: use following theorem: H is a subgroup of the group G if and only if it is nonempty and closed under products and inverses. (These two conditions can be combined into one equivalent condition: whenever a and b are in H, then ab−1 is also in H.) Suppose for any element x(AB) and y(AB) => xA and xB and xA and yB Since A is subgroup of G and xA, yA =>xy-1A Since B is subgroup of G and xB, yB =>xy-1B So xy-1(AB) By above theorem => AB is subgroup of G