Recall that a proper subgroup is called maximal if it is not properly contained in any other proper subgroup. Prove that if is a maximal subgroup of , then either or . Deduce that if is a nonnormal maximal subgroup of a finite group then the number of nonidentity elements that are contained in conjugates of is at most .
Since is a subgroup, we have ; clearly then or . If is not normal, then .
The number of conjugates of is . Now all conjugates of have the same cardinality as , and we will have the largest number of nonidentity elements in the conjugates of precisely when these conjugates intersect trivially. In this case, the number of nonidentity elements in the conjugates of is at most .