2. Are the groups Z/2Z x Z/12Z and Z/4Z x Z/6Z isomorphic? Why or why not?

Find all group homomorphisms from Z₂ XZ₁ to Z₂ XZ₁ 3 4 2 3 Find all group homomorphisms from 7/3 X 7/4 to 1/₂ X 1/3

Consider the group D4 = (a, b) = {e = (1), a, a², a³, b, ab, a²b, a³b} %3D %3D where a = (1 2 3 4) and b = (2 4). Compute (ab)(ab) Compute (ab)(a²b) Compute (ab)(a³b)

he following is a multiplication table for a group G. The product a * b is given in the ble where a is from the leftmost column and b from the top row. Let H e V W X Y 2 = e e V W X W V Y Z X W X Y e W X Z V SOSN e What are the orders of the elements? X 3 Y 2 Y 2 Z X V Y e e X W V W e V Y 3 NS Y ১৯৯ {e, z). Show that this is a subgroup of G. Calculate the right cosets of H. Without calculating the left cosets of H, explain why H is a normal subgroup of G. Construct a multiplication table for the quotient group G/H. Is G/H cyclic? Give reasons. Describe the subgroup Z ≤ G/H generated by the element Hx = G/H. What is the order of (G/H)/Z?

Hello, can you help me out with this problem? Please write a solution on a piece of paper and upload it here.  Introduction of the problem: Let G = ℤ32 and H be the group of G generated by 4 Problem: What is (3 + H)-1 in G/H?

. Prove that the set  S0(2) =  { : z,y  }  forms an abelian group with respect to multiplication.

8 1 C A B (ii), (iii) (i), (ii) (i) (iv) (i), (iv) Suppose f: A B is a non-zero group homomorphism. Which of the following are true? (i): If A = Z36, B = Z6, f(2)=4, then f(10) = 2. (ii): If A = B = Z20, f (7) = 9, then f(1) = 6. (iii): If f is an isomorphism, then ker(f)# {e}. (iv): If A = U10, B = Z₁, f(3) = 2, then ker(f) = {1}.

Which of the following groups are isomorphic? i) Z/75Z ii) Z/3Z e Z/25Z iii) Z/15Z e Z/5Z iv) Z/3Z e Z/5Z OZ/5Z.

Find the order of the element 2+ (6) in the group Z30/(6).

(3) Find the splitting g and UIIC .4 (4) Find the Galois group of the polynomial r* r+1. (5) Find the roots a he polynomia