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Prove that the intersection of any two subgroups of a group ia s again a subgroup?
Let H and I be subgroups of G. A group B is a subgroup of a group A if and only if every member of B is a member of A. Thus, every member of H is a member of G, and every member of I is a member of G. The intersection of two groups A and B is the set of things that are members of both A and B. Hence, if a group C is equivalent to the intersection of A and B, then everything that is a member of both A and B is a member of C, and everything that is a member of C is a member of both A and B. Hence, everything that is a member of the intersection of H and I is a member of H. We established above that every member of H is a member of G. Thus, everything that is a member of the intersection of H and I is a member of G. Recall that A group B is a subgroup of a group A if and only if every member of B is a member of A. Hence, if everything that is a member of the intersection of H and I is a member of G, then the intersection of H and I is a subgroup of G, and so the intersection of H and I is indeed a subgroup of G.
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