If $G$ is a finite group, and $n$ is the least positive integer such that $G$ can be embedded in symmetric group $S\_n$, then, should $G$ necessarily contain a subgroup of index $n$?



Embedding of finite groups in Symmetric Groups
This is not true. The group $S\_3 \times S\_2$ can be embedded in $S\_5$, but not in smaller symmetric groups. 


