The group table of a group G of order 10 is given below. e a b C d f g h i j e e a b C d f 9 h i j a agdhje cib f bb dg fci jeah с chfbegidja dd jce hbfagi ffeig bj a c h d ggcji fa hbde hhieda cbjfg i iba jg hd fe C jfhaidegcb (a) Show that G is cyclic and has d as a generator. (b) Find a subgroup of G of order 5. (c) Write down two isomorphisms from the subgroup in part (b) to (Z5, +5).

Let G= a be a cyclic group of order 35. List all elements having each of the following orders in G. a. 2 b. 5 c. 7 d. 10

Prove that Ca=Ca1, where Ca is the centralizer of a in the group G.

Let G be the group and H the subgroup given in each of the following exercises of Section 4.4. In each case, is H normal in G? Exercise 3 b. Exercise 4 c. Exercise 5 d. Exercise 6 e. Exercise 7 f. Exercise 8 Section 4.4 Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Let H be the subgroup (1),(2,3) of S3. Find the distinct left cosets of H in S3, write out their elements, partition S3 into left cosets of H, and give [S3:H]. Find the distinct right cosets of H in S3, write out their elements, and partition S3 into right cosets of H. In Exercises 7 and 8, let G be the multiplicative group of permutation matrices I3,P3,P32,P1,P4,P2 in Example 6 of Section 3.5 Let H be the subgroup of G given by H=I3,P4={ (100010001),(001010100) }. Find the distinct left cosets of H in G, write out their elements, partition G into left cosets of H, and give [G:H]. Find the distinct right cosets of H in G, write out their elements, and partition G into right cosets of H. Let H be the subgroup of G given by H=I3,P3,P32={ (100010001),(010001100),(001100010) }. Find the distinct left cosets of H in G, write out their elements, partition G into left cosets of H, and give [G:H]. Find the distinct right cosets of H in G, write out their elements, and partition G into right cosets of H.

Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group e,a,b,ab=ba.

Find the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.

The group table of a group G is shown below. e a b e a 9 h e a b a e f C dfgh C dfgh gf d с b h g a d с fehb a d gfe C b a cda h e da bh e f e f g b b h e С с 9 d d h f f b 9 C hd a b c The subset J = {e, b} is a subgroup of G. (a) List all the distinct left cosets and all the distinct right cosets of J in G. (b) State whether J is normal in G, briefly justifying your answer. mb ban (8,3 101]

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?

Part-II: State whether each of the following statements is true or false. Then give an explanation, proof or counter example to justify your answer. Q 9 Let a and b be elements of a group G. If a = a-1 and b = b-1, then be is the inverse of ab. Q 10 Let g and h be elements of a group G such that għ = e. Then hg = e. Q 11 For any two elements r and y in a group G, there is a unique element z in G such that y = xz.

Fill in the blanks in the table below so that it becomes the Cayley table of a group with elements (a, b, c, d}. a b c d a b с d C c d C D