Let G be a p-group and H a nontrivial normal subgroup. Prove that is nontrivial.

Let G act on H by conjugation. Since H is a normal subgroup, this is a well-defined group action since for all .

Let .

.

Therefore we have where .

By the Orbit-Stabilizer Theorem,

Let . By Lagrange’s Theorem, . Since , therefore .

Hence, .

Note that

Therefore . Since , this implies that . Therefore .

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