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Find the sum of the 5th roots of unity?

Which of the following statement are True?1. If ( R, *_1 ,*_2) is a ring ,then psi : R ×R --> psi (r_1 , r_2) = r_1 *_2 r_1 is a binary operation?

Show that d : Q[x] / { 0} --> N union {0}: d(f) = 2 ^(deg f) is a Euclidean valuation on Q [x]?

Check whether f : (4 Z, +) --> (Z_4 , +) : f (4m) = bar m is a group of homomorphism or not.If it is, what does the Fundamental Theorem of Homomorphism gives us in this case? If f is not a Homomorphism, obtain the range of f ?

How many Sylow 5- subgroups, Sylow 3-subgroups and Sylow 2- subgroups can a group of order 200 have? Give reason for your answers?

Let G be a group , H ∆= G and beta <= (G/H) . Let A { x belongs to G | H x belongs to beta } .Show that (1) A <= G (2) H ∆= A (3) beta =(A/H).

For x belongs to G ,define H_x = { g ^(-1) x g | g belongs to G }. Under what condition on x will H_x <= G? Further , if H_x <= G , will H_x ∆= G ? Give reason for your answer?

Which of the following statement are True?1. If G =<x> is of order 25 , then x^(alpha) generates G, where alpha is a factor of 25?

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)

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)