Abstract Algebra Answers

Prove that x * x * x *K* x (n times) = (1+ x) −1 " nÎN n and xÎX .

Define a relation R on Z , by R ={( ,n n + 3 k|)k ∈Z.
Check whether R is an equivalence relation or not. If it is, find all the distinct
equivalence classes. If R is not an equivalence relation, define an equivalence
relation on Z .

Define a relation R on Z , by } R ={( ,n n + 3 k|)k ∈Z.
Check whether R is an equivalence relation or not. If it is, find all the distinct
equivalence classes. If R is not an equivalence relation, define an equivalence
relation on Z . (5)
b) Consider the set }1 X = R {\ − . Define ∗ on X by
X x x x x x x x , x 1
∗ 2 = 1 + 2 + 1 2∀ 1 2 ∈ .
i) Check whether ) ( ,X ∗ is a group or not.
ii) Prove that x ∗ x ∗ x ∗K∗ x (n times) = 1( + )x −1 ∀ n∈N
n
and x ∈X

Let G be a group of order n ≥ 2 , with only two subgroups -{e} and itself. Find a
minimal generating set for G . Also, find out whether n is a prime or a composite
number, or can be either.

Explicitly give the elements and structure of the group Sn/An, n≥5.

Let R be a commutative ring with unity and r ∈ R . Prove that R[x] / (x-r) ≅ Rusing the Fundamental Theorem of Homomorphism.
Hence show that R[x,y]/(y-r)≅ R[x].

Let D = {f (x,y) + g(x,y)i | f,g ∈Z [x,y] }⊆ C [x,y] . Check whether D is a UFD
or not.

Let R = Z[ √2 ] and M = {a + b √2 ∈ R | 5|a and 5|b}
i) Show that M is an ideal of R .
ii) Show that if 5|a or 5|b , then (a^2 + b^2), for a,b∈Z
iii) Hence show that if N is an ideal of R properly containing M , then N = R .
iv) Show that R/M is a field, and give two distinct non-zero elements of this
field.

Define a relation R on Z , by R = {( n, n + 3 k) | k ∈Z} .
Check whether R is an equivalence relation or not. If it is, find all the distinct
equivalence classes. If R is not an equivalence relation, define an equivalence
relation on Z .