  We prove that torsion subgroups of groups defined by C(6), C(4)-T(4) or
C(3)-T(6) small cancellation presentations are finite cyclic groups. This
follows from a more general result on the existence of fixed points for locally
elliptic (every element fixes a point) actions of groups on simply connected
small cancellation complexes. We present an application concerning automatic
continuity. We observe that simply connected C(3)-T(6) complexes may be
equipped with a CAT(0) metric. This allows us to get stronger results on
locally elliptic actions in that case. It also implies that the Tits
Alternative holds for groups acting on simply connected C(3)-T(6) small
cancellation complexes with a bound on the order of cell stabilisers.
