Question

Let G be a group with subgroups H and K. (a) Prove that H ∩ K...

Let G be a group with subgroups H and K.

(a) Prove that H ∩ K must be a subgroup of G.

(b) Give an example to show that H ∪ K is not necessarily a subgroup of G.

Note: Your answer to part (a) should be a general proof that the set H ∩ K is closed under the operation of G, includes the identity element of G, and contains the inverse in G of each of its elements, provided that H and K are subgroups of G. Remember that what needs to be done to establish that some element x belongs to H ∩ K is to show that x belongs to H and x belongs to K. Your answer to part (b) should present a specific group with a pair of specific subgroups, whose union is not a group for some demonstrated reason. Examples of pairs of subgroups like this are plentiful, once you give yourself a particular group in which to work.

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