We investigate the properties of torsion groups and their essential
extensions in the category AbShL of Abellan groups in a topos of sheaves on a
locale. We show that every torsion group is a direct sum of its p-primary components
and for a torsion group A, the group [A,B] is reduced for any Bε
AbShL.. We give an
example to show that in AbShL the torsion subgroup of an injective group need not be
injective. Further we prove that if the locale is Boolean or finite then essential
extensions of torsion groups are torsion. Finally we show that for a first countable
hausdorff space X essential extensions of torsion groups in AbSh0(X) are torsion iff X
is discrete.
