Question

In: Advanced Math

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

Solutions

Expert Solution


Related Solutions

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.
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 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).
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)
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 P be a finite p-group. Show that Φ(P) is the unique normal subgroup of P...
Let P be a finite p-group. Show that Φ(P) is the unique normal subgroup of P minimal such that the corresponding factor group is elementry abelian
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.
show that if H is a p sylow subgroup of a finite group G then for...
show that if H is a p sylow subgroup of a finite group G then for an arbitrary x in G x^-1 H x is also a p sylow subgroup of G
Show that M is a subgroup N; N is a subgroup D4, but that M is not a subgroup of D4
D4 = {(1),(1, 2, 3, 4),(1, 3)(2, 4),(1, 4, 3, 2),(1, 2)(3, 4),(1, 4)(2, 3),(2, 4),(1, 3)} M = {(1),(1, 4)(2, 3)} N = {(1),(1, 4)(2, 3),(1, 3)(2, 4),(1, 2)(3, 4)} Show that M is a subgroup N; N is a subgroup D4, but that M is not a subgroup of D4
Let D3 be the symmetry group of an equilateral triangle. Show that the subgroup H ⊂...
Let D3 be the symmetry group of an equilateral triangle. Show that the subgroup H ⊂ D3 consisting of those symmetries which are rotations is a normal subgroup.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT