Assume is a proper subgroup of the finite group . Prove that ; i.e., is not the union of the conjugates of any proper subgroup.
There exists a maximal subgroup containing . If is normal in , then . If is not normal, we still have . By the previous exercise, contains at most nonidentity elements. Thus , since .
In particular, because is finite, . Thus is not the union of all conjugates of any proper subgroup.