Question

the intersection of any two subgroups of a group is ​

Answer 1

The lattice of subgroups is the set of all subgroups partially ordered by inclusion. The meet operation in this lattice is the intersection of subgroups, and the join operation is the join of subgroups (or, the subgroup generated). The biggest element is the whole group and the smallest element is the trivial subgroup.

Answer 2

Answer:

H is not empty: If e is the neutral element of G, then e∈H1 and e∈H2, because these are subgroups. Hence also e∈H and H≠∅. If a,b∈H, then ab−1∈H: Indeed, a,b∈H implies a,b∈H1 as H⊆H1, hence ab−1∈H1 because H1 is a subgroup. Similarly, ab−1∈H2 and hence ab−1∈H....