The number of elements in conjugates of a maximal subgroup is bounded

Recall that a proper subgroup is called maximal if it is not properly contained in any other proper subgroup. Prove that if M is a maximal subgroup of G, then either N_G(M) = M or N_G(M) = G. Deduce that if M is a nonnormal maximal subgroup of a finite group G then the number of nonidentity elements that are contained in conjugates of M is at most (|M| - 1) \cdot [G : M].


Since M is a subgroup, we have M \leq N_G(M) \leq G; clearly then N_G(M) = M or N_G(M) = G. If M is not normal, then N_G(M) = M.

The number of conjugates of M is |G \cdot M| = [G : N_G(M)] = [G:M]. Now all conjugates of M have the same cardinality as M, and we will have the largest number of nonidentity elements in the conjugates of M precisely when these conjugates intersect trivially. In this case, the number of nonidentity elements in the conjugates of M is at most (|M|-1) \cdot [G : M].

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