CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-11

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$S_{3}$ has trivial center, so the finite $p$ -group hypothesis in the nontrivial-center theorem is necessary

Statement refuted

False claim. Every nontrivial finite group has nontrivial center.

Facts & Assumptions

Given: The symmetric group $S_{3}$ .

[L1]

Every nontrivial finite $p$ -group has nontrivial center (Every nontrivial finite $p$ -group has nontrivial center, in fact $p$ divides $∣ Z (P) ∣$ ).

[L2]

The conjugacy classes of $S_{3}$ have sizes $1$ , $2$ , and $3$ , with only the identity class a singleton (The class equation of $S_{3}$ is $6 = 1 + 2 + 3$ ).

[L3]

A central element commutes with every group element (The center $Z (G)$ of a group).

Counterexample

technique · direct

1.1

An element is central exactly when its conjugacy class is a singleton. By [L2], the only such element of $S_{3}$ is $e$ , so $Z (S_{3}) = {e}$ .

L2L3

2.1

The group $S_{3}$ is nontrivial and finite but has trivial center. Thus the false claim fails, and [L1] shows precisely why the $p$ -group hypothesis cannot be omitted.

step 1.1L1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

T. W. Judson, Abstract Algebra: Theory and Applications, 14.2 (standard reference, not scraped)

S3S3​ has trivial center, so the finite pp-group hypothesis in the nontrivial-center theorem is necessary