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.

Expert Solution

Colby Messinger answered 2 years ago

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 N be a normal subgroup of the group G. (a) Show that every inner automorphism...

Let N be a normal subgroup of the group G. (a) Show that every inner automorphism of G defines an automorphism of N. (b) Give an example of a group G with a normal subgroup N and an automorphism of N that is not defined by an inner automorphism of G

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 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...

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)

Let (G,+) be an abelian group and U a subgroup of G. Prove that G is...

Let (G,+) be an abelian group and U a subgroup of G. Prove that G is the direct product of U and V (where V a subgroup of G) if only if there is a homomorphism f : G → U with f|U = IdU

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.

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)

Given a group G with a subgroup H, define a binary relation on G by a...

Given a group G with a subgroup H, define a binary relation on G by a ∼ b if and only if ba^(-1)∈ H. (a) (5 points) Prove that ∼ is an equivalence relation. (b) (5 points) For each a ∈ G denote by [a] the equivalence class of a and prove that [a] = Ha = {ha | h ∈ H}. A set of the form Ha, for some a ∈ G, is called a right coset of H...

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

Question