Let G = G1 x G2. Let H = {(x1, e2) : x1 ∈ G1} and K = {(e1, x2) : x2 ∈ G2}.
(a) Prove H ≤ G and K ≤ G.
(b) Prove that HK = KH = G
(c) Prove that H ∩ K = {(e1, e2)}
(d) Show that G/H is isomorphic to G2 and G/K is isomorphic to G1.
I have no idea where to start. I understand the rules and tests to use but I do not now how to apply it to this question.
a) We clearly have closure since (x,e2)(x',e2) = (x1x2,e2e2) = (x1x2, e2), and we also have identity since in particular x=e1 is in our set, and we also have inverses since all the elements of G1 have inverses (i.e we have a natural isomorphism of G1 onto H). Hence, H is a subgroup. A symmetrical argument works to show that K is also a subgroup.
b) Well, HK = {(x,e2)(e1,y)}. But clearly all elements of G are of the form (x, y)=(x,e2)(e1,y). Again, symmetrically KH= G.
c) Well, this intersection is the set of elements (x,y) where x=e1 and y=e2 simply by definition of set intersection, so there really isn't much to do here.
d) G/H = {(x,y) + H} = {(e1, y) + H} because (x, e1) is in H, and so that coordinate won't affect the cosets. Argue symmetrically for K.