 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2012/b/b2/f/solution

Write

∣ G ∣ = p^{n}

, where

n \geq 1

. By part (c), each conjugacy-class size divides

p^{n}

and hence is a power of

p

. Every nonsingleton class consequently has size divisible by

p

∣ Z (G) ∣ \equiv ∣ G ∣ \equiv 0 (mod p) .

The identity element lies in the centre, so the centre is nonempty. Its size is a positive multiple of

p

, and therefore

∣ Z (G) ∣ \geq p \geq 2.

This proves that a nontrivial finite p-group has a nontrivial centre.

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