no. (Q-{0},*) is infinite, but its torsion group is not empty (-1 has order 2).

if G is finite, every element has to have finite order, so for a finite group, Tor(G) = G.So suppose G is finite.

this is really quite simple. suppose a,b are of finite order m and n, respectively.

then because G is abelian

so ab must be of finite order.

for the non-abelian case, the only example that comes to mind is the free group on two generators a,b subject to the relations .

clearly ab is of infinite order.