Suppose G is a group of infinite order and H is a subgroup of G. What are the possibilities for |G/H|?
|G/H| can be finite and |G/H| can be infinite; both possibilities
occur
|G/H| can only be infinite
|G/H| can only be finite
Indeed, if G is the additive group of Z,
then H = {0} has infinitely many left cosets
and H = 2Z has finitely many.
No, for instance
H = 2Z has finitely many left cosets in
the additive group of Z.
No, for instance
H = {0} has infinitely many left cosets in
the additive group of Z.