Abstract Algebra Answers

Factorise 10 in two ways in Z[
√
−6]. Hence, show that Z[
√
−6] is not a UFD.

(a) Show that <x > is not a maximal ideal in z[x].

(b)List all the subgroups of Z18, along with 3 their generators.

(c)Let H=< (1 2) > and k = < (1 2 3) > be subgroups of S3. Show that S3 = Hk. Is S3 an internal direct product of H and k ? Justify your answer.

(d)Check whether or not { (2, 5), (1, 3), (5, 2), (3, 1) is an equivalence relation on { 1, 2, 3, 5 }.

find the perimeter of a triangular garden that measures 10ft x 10ft by (2x+3)feet. Use P =a+b+c

A rug is cut to fit in a room so that a border of consistent width is left on all four sides. If the room is 10 feet by 18 feet and the area of the rug is 84 feet, how wide will the border be?

what is product of 6 2/3 x 9/40

a) Let
S =

a 0
0 b

a; b 2 Z

:
i) Check that S is a subring of M2
(R) and it is a commutative ring with identity.
ii) Is S an ideal of M2
(R)? Justify your answer.
iii) Is S an integral domain? Justify your answer.
iv) Find all the units of the ring S.
v) Check whether
I =

a 0
0 b

a; b 2 Z; 2 j a

:
is an ideal of S.
vi) Show that S ' Z Z where the addition and multiplication operations are componentwise
addition and multiplication.
b) Let G = S
4, H = A4
and K = f1; (1 2)(3 4); (1 3)(2 4); (1 4)(2 3)g.
i) Check that H=K = h(1 2 3)Hi
ii) Check that K is normal in H.(Hint: For each h 2 H,h 62 K, check that hK = Kh.)
iii) Check whether (1 2 3 4))H is the inverse of (1 3 4 2)H in the group S
4
=H.

Let D12 = h

x; y x
2
= e; y
6
= e; xy = y
1
x i.
a) Which of the following subsets are subgroups of D12? Justify your answer.
i) fx; y ; xy ; y
2
; y
3
; eg ii) fxy ; xy
2
; y
2
; eg iii) fx; y
3
; xy
3
; eg (4)
b) Find the order of y
2
. Is the subgroup hy
2
i normal? Justify your answer.
c) Let
D2n = h

x; yj x
2
= e; y
n
= e; xy = y
1
x i
Prove the relation
x
i
y
j
x
k
y
l
=
(
x
i
y
j+l
if k is even
x
i+k
y
l j
if k is odd.
Further, find all the elements of order 2 in D12.
d) Find two different Sylow 2-subgroups of D12.

The map f : R[x] -> M(subscript3)(R) is defined by
f ( a(subscript 0) + a(subscript 1)x + a(subscript 2)x(power 2) + ....... + a(subscript n)x(power n)
_ _
| a(subscript 0) a(subscript 1) a(subscript 2) |
= | 0 a(subscript 0) a (subscript 1) |
|_ 0 0 a(subscript 0) _|
Show that f is a group homomorphism.Determine ker(f) also.

Let D(subscript12) = ({x,y : x^2 = e ; y^6 = e ; xy =(y^-1) x})
a) Which of the following subsets are subgroups of D(subscript12) ? Justify your answer.
i) {x,y,xy,y^2,y^3,e} ii) {xy,xy^2,y^2,e} iii) {x,y^3,xy^3,e}
b) Find the order of y^2. Is the subgroup (y^2) normal ? Justify your answer.
c) Let D(subscript2n) = ({x,y : x^2 = e ; y^n = e ; xy =(y^-1) x})
Prove the relation
{ x^i*y^(j+l) if k is even
X^i*y^j*x^k*y^l = {
{ x^(i+k)*y^(l-j) if k is odd
Further, find all the elements of order 2 in D(subscript12) .
D) Find two different Sylow 2-subgroups of D(subscript12) .

Let D(subscript12) =({x,y : x^2 = e ; y^6 = e ; xy =(y^-1) x})
a) Which of the following subsets are subgroups of D(subscript12) ? Justify your answer.
i) {x,y,xy,y^2,y^3,e} ii) {xy,xy^2,y^2,e} iii) {x,y^3,xy^3,e}