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

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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) About Me  (Show Source):
You can put this solution on YOUR website!
Let C = set of all alpha in G such that %28xy%29%2Aalpha+=+alpha%2A%28yx%29.
Suppose that C', the complement of C, is non-empty.
Let gamma, beta ∈ C'.
Then for any element z of G,
gamma%2Az+=+z%2Abeta ==> gamma=beta.
==> C' = { beta }, a set with the single element beta.
Since beta ∈ C', beta%2A%28yx%29 ∉ C.
This fact forces %28xy%29%2Abeta+=+beta%2A%28yx%29
==> beta ∈ C. Contradiction.
Hence C = ∅, and so for ANY w ∈ G, (xy)w = w(yx)
==> xy = yx for all x,y.
Therefore G is abelian.