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
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
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