Suppose $X$ a infinite set and $S_X$ is the permutation group of $X$. Prove that any proper subgroup of $S_X$ has infinite index.
The nontrivial normal subgroups of the symmetric group on an infinite set $X$ consist of the finitary alternating and symmetric groups, together with the groups whose support is bounded above by a fixed infinite cardinality. (See, for example, Dixon & Mortimer, Permutation Groups, Thereom 8.1A.) So all proper normal subgroups have infinite index and hence the same applies to all proper subgroups.