Find a subgroup G in symmetric (permutation) group Sn such that (1) n = 4 and...

Find a subgroup G in symmetric (permutation) group Sn such that

(1) n = 4 and G is abelian noncyclic group

(2) n = 8 and G is dihedral group.

Expert Solution

Colby Messinger answered 2 years ago

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)

1) What permutation group is pentagon D5 a subgroup of ?why? 2) What is a generating...

1) What permutation group is pentagon D5 a subgroup of ?why? 2) What is a generating set of pentagon D5? Why? Is it a minimal generating set?Why or why not? 3) What is not a generating set of pentagon D5? Why?

Give an example of a nonabelian group G of order n and a subgroup H of...

Give an example of a nonabelian group G of order n and a subgroup H of order k. Then list all of the cosets of G/H. where n = 24 and k = 3.

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 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

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)

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

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