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.

## Comments

in the second to last sentence, (the middle term of the inequality), do you mean

lGl – [G:M] +1 instead of [g:M]?

Yes.🙂 Thanks!

The phrase “because G is finite” seems unnecessary. The finiteness was used for the formula in the previous exercise.