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 .
Therefore we have where .
By the Orbit-Stabilizer Theorem,
Let . By Lagrange’s Theorem, . Since , therefore .
Therefore . Since , this implies that . Therefore .
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