Recall that we proved the following when we discussed product groups: Proposition 2.11.4: Let H and K be subgroups of a group G, and let f:HXK → G be the multiplication map, defined by f(h, k) = hk. Its image is the set HK = {hkhe H, EK}. Then: (i) f is injective if and only if H K = {1}.
(ii) f is a homomorphism from the product group H x K to G if and only if elements of K commute with elements of H: hk = kh.
(iii) If H is a normal subgroup of G, then HK is a subgroup of G.
(iv) f is an isomorphism from the product group H x K to G if and only if H K = {1}, HK = G, and also H and K are normal subgroups of G.
Keep the above in mind for the following problems:
(a) Let G be a non-cyclic group of order 21. Compute n7(G), the number of Sylow 7- subgroups of G, and nz(G), the number of Sylow 3-subgroups of G.
(b) Let G be a group of order 175. Prove that G is abelian.