2.4 Abstract Algebra 91(ii) (identity element) there ise∈Gsuch that for anyx∈G,ex=xe=x;
(iii) (existence of the inverse) for everyx ∈Gthere isx−^1 ∈Gsuch thatxx−^1 =
x−^1 x=e.
But Cayley observed the following fact.
Theorem.Any group is a group of transformations.
Proof.Indeed, any groupGacts on itself on the left. Specifically,x ∈Gacts as a
transformation ofGbyy→xy,y∈G.
A groupGis called Abelian (after N. Abel) if the operation is commutative, that is,
ifxy=yxfor allx, y∈G. An example of an Abelian group is the Klein four-group,
introduced abstractly asK={a, b, c, e|a^2 =b^2 =c^2 =e, ab=c, ac=b, bc=a},
or concretely as the group of the symmetries of a rectangle (depicted in Figure 14).
b
a
c
Figure 14A group is called cyclic if it is generated by a single element, that is, if it consists of
the identity element and the powers of some element.
Let us turn to problems and start with one published by L. Daia in theMathematics
Gazette, Bucharest.
Example.A certain multiplicative operation on a nonempty setGis associative and
allows cancellations on the left, and there existsa∈Gsuch thatx^3 =axafor allx∈G.
Prove thatGendowed with this operation is an Abelian group.
Solution.Replacingxbyaxin the given relation, we obtainaxaxax=a^2 xa. Cancelling
aon the left, we obtainx(axa)x=axa. Becauseaxa=x^3 , it follows thatx^5 =x^3 ,
and cancelling anx^2 , we obtain
x^3 =x for allx∈G.In particular,a^3 =a, and hencea^3 x=axfor allx∈G. Cancelaon the left to find that
a^2 x=x for allx∈G.