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Define a relation R on Z by R={(n,n+3k)| 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

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

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

Consider the ring Homomorphism phi : Z [x] --> Z/(3) : phi [summation (n) (i=0) of a_i x^i ]= a_0 bar .Show that Ker phi = (x,3) . What does the Fundamental Theorem of Homomorphism say in this case ?

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

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 ) .

(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

State 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

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?