Question

In: Advanced Math

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).

Solutions

Expert Solution


Related Solutions

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...
1. Let N be a normal subgroup of G and let H be any subgroup of...
1. Let N be a normal subgroup of G and let H be any subgroup of G. Let HN = {hn|h ∈ H,n ∈ N}. Show that HN is a subgroup of G, and is the smallest subgroup containing both N and H.
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.
G is a group and H is a normal subgroup of G. List the elements of...
G is a group and H is a normal subgroup of G. List the elements of G/H and then write the table of G/H. 1. G=Z10, H= {0,5}. (Explain why G/H is congruent to Z5) 2. G=S4 and H= {e, (12)(34), (13)(24), (14)(23)
(a) Suppose K is a subgroup of H, and H is a subgroup of G. If...
(a) Suppose K is a subgroup of H, and H is a subgroup of G. If |K|= 20 and |G| = 600, what are the possible values for |H|? (b) Determine the number of elements of order 15 in Z30 Z24.
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...
1. Let G be the symmetry group of a square and let H be the subgroup...
1. Let G be the symmetry group of a square and let H be the subgroup generated by a rotation by 180 degrees. Find all left H-cosets.
Let G be a group and let N ≤ G be a normal subgroup. (i) Define...
Let G be a group and let N ≤ G be a normal subgroup. (i) Define the factor group G/N and show that G/N is a group. (ii) Let G = S4, N = K4 = h(1, 2)(3, 4),(1, 3)(2, 4)i ≤ S4. Show that N is a normal subgroup of G and write out the set of cosets G/N.
4.- Show the solution: a.- Let G be a group, H a subgroup of G and...
4.- Show the solution: a.- Let G be a group, H a subgroup of G and a∈G. Prove that the coset aH has the same number of elements as H. b.- Prove that if G is a finite group and a∈G, then |a| divides |G|. Moreover, if |G| is prime then G is cyclic. c.- Prove that every group is isomorphic to a group of permutations. SUBJECT: Abstract Algebra (18,19,20)
Prove that a subgroup H of a group G is normal if and only if gHg−1...
Prove that a subgroup H of a group G is normal if and only if gHg−1 =H for all g∈G
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT