"If G is a subgroup of the symmetric group S_n which contains an odd permutation, then G is a subgroup of index 2"
"If G is a group of order (2^k)m, m odd, and the Sylow 2-subgroups of G are cyclic, then G has a normal subgroup of order m. Hint: Let G act on G by left multiplication and apply the preceding problem."
I posted these questions together since they seem to be very similar.