abstract algebra Consider the group GU2 which cuteiets of t

abstract algebra

) Consider the group GU(2) which cuteiets of the 12 elements These elements have orden 1.6. Ial, 3, 6, 2.M.6,6. 6,3,1 respectively (a) What are the elements a, b and c in the list above? (b) Determine, th\" orders lu],/and iejof thme element . ic) Find subgroups H and K ol G o orders 4 and 3 respectively such th (d] Which of the following is true? Explain why. (i) (e (Z/42) × (Z/32)

Solution

group U(28) has all the elements (less than 28)coprime to 28

a)

so a = 5

b = 15

c = 27

order of 5 = 6

5^2 = 25 , 5^3 = 125 = 13 mod 28 , 5^4 = 9 mod 28 ,5^5 = 17 mod 28 5^6 = 1 mod 28

order of 15 = 2

15^2 = 225 = 1 mod 28

order of 27 = 2

27^2 = 1 mod 28

c) H = {1, 13, 15,27} all elements of order 2 and unit element

K = {1,9,25} all elements of order 4 and unit element

H X K will have unit element (1,1) , three elemets of order 2 (13,1) , (15,1) , (27,1), two elements of order 3 (1,9) , (1,25) and the rest of elements of order 6. Hence this is isomorphic to G

d) G is isomorphic to H X K

but H is not isomorphic to Z/4Z since it has an element of order 4 (and hence it is cyclic)

so G is not isomorphic to Z/4Z X Z/3Z

H is isomorphic to Z/2Z X Z/2Z

and K is isomorphic to Z/3Z

hence G is isomorphic to Z/2Z X Z/2Z X Z/3Z