Let G be a group and let H,K be normal subgroups of G such that H∩K = {e} and that G = {hk|h ∈ H,k ∈ K}. (1)Prove that for every h∈H, k∈K we have kh(k^-1)(h^−1) = e in G. (2) Prove that the group G is isomorphic to H × K. Hint: For (2), consider the map φ : H ×K → G, defined as φ(h,k) = hk, whereh ∈ H,k ∈ K.
(8) (5 Points Total) Let (G, ) be a group and H and K are subgroups of G, such that H¢K and K¢ H. Is HuK is a subgroup of G? Prove or disprove. (Note: it is not enough to say Yes or No) Answer:
Problem 3. Subgroups of quotient groups. Let G be a group and let H<G be a normal subgroup. Let K be a subgroup of G that contains H. (1) Show that there is a well-defined injective homomorphism i: K/ H G /H given by i(kH) = kH. By abuse of notation, we regard K/H as being the subgroup Imi < G/H consisting of all cosets of the form KH with k EK. (2) Show that every subgroup of G/H is...
proof please 51. Let H and K be subgroups of an abelian group G of orders n and m respectively. Show that if H K = {e}, then HK = {hkh e H and ke K} is a subgroup of G of order nm.
thanks 9. (10 ) Suppose that H and K are distinct subgroups of G of index 2. Prove that HnK is a normal subgroup of G of index 4 and that G/(Hn K) is not cyclic. (Hint. Use the 2nd Isomorphism Theorem) 9. (10 ) Suppose that H and K are distinct subgroups of G of index 2. Prove that HnK is a normal subgroup of G of index 4 and that G/(Hn K) is not cyclic. (Hint. Use the...
(a) Show that if H and K are subgroups of an abelian group G, then HK = {hk|he H, k E K} is a subgroup of G (b) Show that if H and K are normal subgroups of a group G, then H N K is a normal subgroup of G
(4)(20 points) (a) Show that if H and K are subgroups of an abelian group G, then HK = {hk|he H, ke K}is a subgroup of G (b) Show that if Hand K are normal subgroups of a group G, then H N K is a normal subgroup of G
(4)(20 points) (a) Show that if H and K are subgroups of an abelian group G, then HK = {hk|he H, KE K} is a subgroup of G. (b) Show that if H and K are normal subgroups of a group G, then HK is a normal subgroup of G
19. Let G be a group with no nontrivial proper subgroups. (a) Show that G must be cyclic. (b) What can you say about the order of G?
question for 10. (16M) Let H and K be subgroups of G. Define HK = {hk |h E H,kE K}. Suppose K is normal in G. Prove (a) HK is a subgroup of G. (b) HnK is a normal subgroup of H; K is a normal subgroup of the subgroup H K. HK K H (c) HnK (16M) Let H and K be subgroups of G. Define HK = {hk |h E H,kE K}. Suppose K is normal in G....