CounterexampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 44 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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

S3S_3S3​ has trivial center, so the finite ppp-group hypothesis in the nontrivial-center theorem is necessary