Perhaps the following may be of some help, but it requires some knowedge of group theory: if we look at the modular

group

, then

the free product of

the cyclic groups of order 2 and 3, and by a well-known theorem in such a free product an element has finite order iff it

conjugate to a finite order of one of the factors...

This means that any element of order 3 has to be conjugate in the free product to one of the two non trivial elements of

the factor

. Projecting this idea back to the original group we can get the result.

Tonio