3 Clarified a point.

The bijection definition is fine, although it's not a very nice group. One might also consider the group generated by all transpositions on {1, 2, ...}, which is the subgroup of all bijections that fix all but finitely many elements, and this group is likely to be much nicer; it's countable, for one thing.

Edit: I guess it's worth noting that as far as I can tell the term infinite symmetric group is used by mathematicians to refer to the subgroup I described.

2 Clarified a point.

The bijection definition is fine. One might also consider the group generated by all transpositions on {1, 2, ...}, which is the subgroup of all bijections that fix all but finitely many elements, and this group is likely to be much nicer; it's countable, for one thing.

Edit: I guess it's worth noting that as far as I can tell the term infinite symmetric group is used by mathematicians to refer to the subgroup I described.

1

The bijection definition is fine. One might also consider the group generated by all transpositions on {1, 2, ...}, which is the subgroup of all bijections that fix all but finitely many elements, and this group is likely to be much nicer; it's countable, for one thing.