Which of the following are abelian groups? Why? 1) The set of all pairs (x,y) of

Question

Which of the following are abelian groups? Why? 1) The set of all pairs (x,y) of real numbers, under teoperation (x*y)*(z,w) = (x+z,y-w) 2) The set of all pairs (x,y) or real numbers such thaty does not equal 0, under the operation (x,y) * (z,w) =(x+y,yw) Which of the following are abelian groups? Why? 1) The set of all pairs (x,y) of real numbers, under teoperation (x*y)*(z,w) = (x+z,y-w) 2) The set of all pairs (x,y) or real numbers such thaty does not equal 0, under the operation (x,y) * (z,w) =(x+y,yw)

Explanation / Answer

1) This is NOT an abelian group since       (x,y)*(z,w) = (x+z,y-w)and       (z,w)*(x,y) =(z+x,w-y) Thus, (x,y)*(z,w) is not equal to (z,w)*(x,y) 2) This is an abelian group. In your problem youwrote that (x,y)*(z,w) = (x+y, yw) but I'm not sure but did youmean (x,y)*(z,w) = (x+z, yw)? I assumed you did. Ifnot, proceed to number 3.       (x,y)*(z,w) = (x+z,yw)= (z+x,wy) since elements (wy = yw) of a group areassociative. And (z+x,wy) = (z,w)*(x,y). Thus,(x,y)*(z,w)=(z,w)*(x,y) 3) This is definitely NOT abelian.    (x,y)*(z,w) = (x+y, yw), but    (z,w)*(x,y) = (z+w, wy) Thus, (x,y)*(z,w) is not equal to (z,w)*(x,y).

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