Honestly, I have been looking for an a finite Quasicommutative semigroup by surfing the web, but I could't. May I ask here to give me an example for such this kind of semigroup. I tried to built one of them by using GAP, but it fails. Thank you so much. I am new to it.

The simple examples are Hamiltonian groups. Then you can construct Clifford semigroups from them. 


The following is (for me) a bit much for a series of comments. EDIT: It is also not appropriate as an answer. The fun begins when I alternate quantifiers and turn the intended property into a semigroup identity. In an actual quasicommutative semigroup, $ab=b^3a$ would not say anything about b having torsion. Even in the finite semigroup case, there is no justification for saying that all nontrivial powers are rth powers. I leave the comedy of inferences for those who like to see how easy and wrong it can be to deduce something from a single conflation of (for all)(exists) and (exists)(for all). END EDIT. It might interest one who wants to construct an example. The propety as described by Martin Brandenburg is a semigroup identity that implies $b^2=b^{r+1}$. Further, one finds that for $i>1$, $b^i=c^r$ for some $c$ and that $r$th powers commute with everything in the semigroup. An attempt at a reduced form for words on k letters in such a semigroup (this part needs checking before taking seriously) is something like wp where w is a squarefree word (so w does not have abcabc or similar subwords) on some subset S of the k letters and p is possibly the empty word and otherwise is an $r$th power involving only powers of the letters not in S. The subsemigroup of square or higher powers will be commutative. If the above is correct, I see finite noncommutative examples looking like a small set of extra elements adjoined to a commutative semigroup, where the square of any extra element lies in the commutative semigroup. Not being a semigroup theorist, I sympathize with the original poster and his/her plight. Gerhard "Ask Me About System Design" Paseman, 2013.02.05 

