Let A and B be groups, and consider the product group G=A x B. (a) Prove...

Let A and B be groups, and consider the product group G=A x B.

(a) Prove that N={(e_a,b) E A x B| b E B} is a subgroup.

(b) Prove that N is isomorphic to B

(d) Prove that G|N is isomorphic to A

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Let G be a group. For each x ∈ G and a,b ∈ Z+ a) prove...

Let G be a group. For each x ∈ G and a,b ∈ Z+ a) prove that xa+b = xaxb b) prove that (xa)-1 = x-a c) establish part a) for arbitrary integers a and b in Z (positive, negative or zero)

Let G be a group,a;b are elements of G and m;n are elements of Z. Prove...

Let G be a group,a;b are elements of G and m;n are elements of Z. Prove (a). (a^m)(a^n)=a^(m+n) (b). (a^m)^n=a^(mn)

Let G and H be groups. Define the direct product G X H = {(x,y) |...

Let G and H be groups. Define the direct product G X H = {(x,y) | x ∈ G and y ∈H } Prove that G X H itself is a group.

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 cyclic group generated by an element a. a) Prove that if an...

Let G be a cyclic group generated by an element a. a) Prove that if an = e for some n ∈ Z, then G is finite. b) Prove that if G is an infinite cyclic group then it contains no nontrivial finite subgroups. (Hint: use part (a))

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.

Prove 1. Let f : A→ B and g : B → C . If g...

Prove 1. Let f : A→ B and g : B → C . If g 。 f is one-to-one, then f is one-to-one. 2. Equivalence of sets is an equivalence relation (you may use other theorems without stating them for this one).

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.

Let G be a group. (consider the following parts that go together): (1) Prove that (a-1ba)n...

Let G be a group. (consider the following parts that go together): (1) Prove that (a-1ba)n = a-1bna for any a,b in G, and any integer n. (2) Prove that |xax-1| = |a| for any a, x in G. (3) Prove: If a is the only element of order two in G, then a lies in Z(G) where Z is the center of the group, G.

Let G be a nontrivial nilpotent group. Prove that G has nontrivial center.

Subjects