SOLUTION: G is a finite semigroup such that for each x, y, z, if xy = yz, then x = z. Prove that G is abelian.

Question 1028299: G is a finite semigroup such that for each x, y, z, if xy = yz, then x = z.
Prove that G is abelian.

Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
Let C = set of all in G such that .
Suppose that C', the complement of C, is non-empty.
Let , ∈ C'.
Then for any element z of G,
==> .
==> C' = { }, a set with the single element .
Since ∈ C', ∉ C.
This fact forces
==> ∈ C. Contradiction.
Hence C = ∅, and so for ANY w ∈ G, (xy)w = w(yx)
==> xy = yx for all x,y.
Therefore G is abelian.

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