Asking about a quasicommutative semigroup 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.
 A: The simple examples are Hamiltonian groups. Then you can construct Clifford semigroups from them.
A: 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
