Abstract Algebra Answers

If σ is an even permutation, then σ ² = I . Write given statement is true or false, justify your answer.

Let S be a nonempty subset of plane 2 \ , it is known that every point ( , ) x y in S
satisfies “if x > 0 , then y > 0 ”. Consider the following properties possibly satisfied
by points( , ) x y in S :
(I) If x ≤ 0 , then y ≤ 0 .
(II) If y ≤ 0 , then x ≤ 0 .
(III) If y > 0 , then x > 0 .
Which of the above properties will have to be satisfied by all points( , ) x y in S?
(a) (II) only
(b) (III) only
(c) (I) and (II)
(d) (I) and (III)
(e) (II) and (III)

The set of discontinuous functions from [1,0] to R form a ring with respect to pointwise addition and multiplication. Is this statement true or false, justify your answer.

The characteristic of a finite field is zero. Is this statement true or false, justify your answer.

If a ring has a unit, then it has only one unit. Is this statement true or false ,justify your answer.

If R is a ring and I is an ideal of R , then xr = rx ∀ x ∈ I and r ∈ R . Is this statement true or false, justify your answer .

Consider the set X=R\{-1}. Define * on X by
x1 * x2= x1+ x2+ x1x2 ¥ x1,x2 € X.
i) check whether (X,*) is a group or not.
ii) Prove that x*x*x*....*x (n times)=(1+x)n -1¥n€N and x€X

Give an example, with justification, of a commutative subgroup of a non commutative
group.

Q. Show that similar matrices have same eigen values.

there exists a non-cyclic group in which every proper subgroup is cyclic.true or false prove