Prove that the intersection of two subgroups is a subgroup.
Let A, B are two subgroup of group G, prove that AB 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(AB) and y(AB) => xA and xB and xA and yB
Since A is subgroup of G and xA, yA =>xy-1A
Since B is subgroup of G and xB, yB =>xy-1B
So xy-1(AB)
By above theorem => AB is subgroup of G