(h, k) E H x K. What is the order Let H and K be finite groups and let G = H x K. Let g of g?

Let G be a group and H<=G suppose that g1, g2 element of G, Show that Hg1 =Hg2 if and only if g1-1H =g1-1H

ew W=W Let w be a group and for each a Ew, let fa: Wow be defined as fa(x) = axa ²¹. Show that fa is an isomorphism.

B3. Show your step-by-step solution and answer.

Q4 True or false ?

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?

true or false  provide solution

新 ole. Let G = A5. We determine the conjugacy classes of G

Let G and H be groups and o : G → H be a homomorphism. c. Define the maps n1 : G x H → G and 72 : G x H → H by 1(9, h) = g and 72(9, h) = h respectively. %3D Show that 71 and n2 are homomorphisms and find their kernels.

Formulate a Cayleyan (multiplication) table and state the reason if why it is a group or not.