Let H and K be subgroups of a group G so that for all h in...

Let H and K be subgroups of a group G so that for all h in H and k in K there is a k' in K with hk = k'h. Proposition 2.3.2 shows that HK is a group. Show that K is a normal subgroup of HK.

Expert Solution

This ia the required proof.For your good understanding,I have proved the result too,which is used.I hope the answer will help you.Expecting a thumbs up if you are satisfied with the work,it will help me a lot.Thank you.

Colby Messinger answered 2 years ago

Let G be a group and H and K be normal subgroups of G. Prove that...

Let G be a group and H and K be normal subgroups of G. Prove that H ∩ K is a normal subgroup of G.

(1) Let G be a group and H, K be subgroups of G. (a) Show that...

(1) Let G be a group and H, K be subgroups of G. (a) Show that if H is a normal subgroup, then HK = {xy|x ? H, y ? K} is a subgroup of G. (b) Show that if H and K are both normal subgroups, then HK is also a normal subgroup. (c) Give an example of subgroups H and K such that HK is not a subgroup of G.

The question is: Let G be a finite group, H, K be normal subgroups of G,...

The question is: Let G be a finite group, H, K be normal subgroups of G, and H∩K is also a normal subgroup of G. Using Homomorphism theorem ( or First Isomorphism theorem) prove that G/(H∩K) is isomorphism to a subgroup of (G/H)×(G/K). And give a example of group G with normal subgroups H and K such that G/(H∩K) ≆ (G/H)×(G/K), with explanation. I was trying to find some solutions for the isomorphism proof part, but they all seems to...

Suppose G is a group and H and H are both subgroups of G. Let HK={hk,...

Suppose G is a group and H and H are both subgroups of G. Let HK={hk, h∈H and k ∈K} a.give a example such that |HK| not equal to |H| |K| b. give a example to show f :HK →H ⨯K given by f(hk) = (h,k) may not be well defined.

Let G be a group and K ⊂ G be a normal subgroup. Let H ⊂...

Let G be a group and K ⊂ G be a normal subgroup. Let H ⊂ G be a subgroup of G such that K ⊂ H Suppose that H is also a normal subgroup of G. (a) Show that H/K ⊂ G/K is a normal subgroup. (b) Show that G/H is isomorphic to (G/K)/(H/K).

Let G, H, K be groups. Prove that if G ≅ H and H ≅ K...

Let G, H, K be groups. Prove that if G ≅ H and H ≅ K then G ≅ K.

Let G be a finite group and H a subgroup of G. Let a be an...

Let G be a finite group and H a subgroup of G. Let a be an element of G and aH = {ah : h is an element of H} be a left coset of H. If b is an element of G as well and the intersection of aH bH is non-empty then aH and bH contain the same number of elements in G. Thus conclude that the number of elements in H, o(H), divides the number of elements...

Let G be an abelian group and K is a subset of G. if K is...

Let G be an abelian group and K is a subset of G. if K is a subgroup of G , show that G is finitely generated if and only if both K and G/K are finitely generated.

Let G be a group acting on a set S, and let H be a group...

Let G be a group acting on a set S, and let H be a group acting on a set T. The product group G × H acts on the disjoint union S ∪ T as follows. For all g ∈ G, h ∈ H, s ∈ S and t ∈ T, (g, h) · s = g · s, (g, h) · t = h · t. (a) Consider the groups G = C4, H = C5, each acting...

Let G = Z4XZ3XZ2 and consider the two cyclic subgroups H = h(0; 1; 1)i and...

Let G = Z4XZ3XZ2 and consider the two cyclic subgroups H = h(0; 1; 1)i and K = h(2; 1; 1)i of G. (a) Find all cosets (along with the elements they contain) to H and K, respectively. (b) Write down Cayley tables for the factor groups G=H and G=K, and classify them according to the Fundamental Theorem of Finite Abelian Groups.

Question