CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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Every group of order $p^2$ , for prime $p$ , is abelian

Statement

If $p$ is prime and $G$ is a group of order $p^2$ , then $G$ is abelian.

Facts & Assumptions

Given: A prime $p$ and a finite group $G$ with $|G|=p^2$ .

[L1]

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

[L2]

If $G/Z(G)$ is cyclic, then $G$ is abelian (If $G/Z(G)$ is cyclic, then $G$ is abelian).

[L3]

For finite $G$ , $|G/Z(G)|=|G|/|Z(G)|$ (If $[G:N]$ is finite then $|G/N|=[G:N]$ ; for finite $G$ this equals $|G|/|N|$ ).

[L4]

A group of prime order is cyclic (A finite group of prime order is cyclic and every nonidentity element generates it).

[L5]

A group of order $p^2$ is a finite $p$ -group (A finite $p$ -group has order $p^n$ for a prime $p$ and some $n\in\mathbb N$ ).

[L6]

Every subgroup of a finite $p$ -group has prime-power order (Every subgroup of a finite $p$ -group has order a power of $p$ ).

[L7]

The center $Z(G)$ is a normal subgroup, hence in particular a subgroup, of $G$ (The center of a group is a normal subgroup).

[L8]

A finite subset with the same cardinality as its ambient finite set is the whole set (A subset of a finite set is finite, with $\lvert B\rvert \le \lvert A\rvert$ , and equality holds if and only if $B = A$ ).

Proof

technique · direct

1.1

By [L1] and [L7], $Z(G)$ is a nontrivial subgroup of $G$ ; [L5] and [L6] therefore show that it has order $p$ or $p^2$ .

L1L5L6L7

2.1

If $|Z(G)|=p^2=|G|$ , then [L8] gives $Z(G)=G$ , so $G$ is abelian. If $|Z(G)|=p$ , then [L3] gives $|G/Z(G)|=p$ , so [L4] makes $G/Z(G)$ cyclic.

step 1.1L3L4L8

3.1

In the second case [L2] makes $G$ abelian, and the first case already did so. Hence every group of order $p^2$ is abelian.

step 2.1L2∎

Depends on

Used by

The square-symmetry group has order 2³ and is nonabelian, so the order-p² theorem does not extend to order p³ Counterexample

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 122 results over 26 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

K. Conrad, Group Actions, Corollary 5.2 (standard reference, not scraped)

Every group of order p2p^2p2, for prime ppp, is abelian