Answer to Question #123770 in Abstract Algebra for Nii Laryea

Question #123770
Determine whether or not the set W = {(x, y, z) ϵ Rᶟ xy = z} is a subgroup of the group
Rᶟ under coordinatewise addition.
1
Expert's answer
2020-06-24T18:44:55-0400

Given W={(x,y,z):xy=z} ...(1)

Recall : A subset is said to be subspace of a vector space if

1) x,y"\\in" W "\\implies" x+y"\\isin" W

2) if x"\\isin" W and a"\\isin" F "\\implies"ax"\\in" W

Now let's check these property for given W

Let x=(x1,y1,z1) and y=(x2,y2,z2) "\\isin" W

Hence , x1y1=z1 and x 2y2=z2

But (x+y)=(x1+x2 ,y1+y2,z1+z2)

So, (x1+x2)(y1+y2)=(z1+z2) is NOT equal to x1y1+x2y2=(z1+z2)

Hence ,(x+y) does not belong to W

Hence W is not a subspace.



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