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✓ 7 results · all verified · 5 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 2 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

Pro P Groups and the P Adic Integers - Examples

1 · Prerequisites

2 · Summary

These examples make the compatible-digit model explicit, show concrete convergence and Cauchy behavior in $Z_{p}$ , identify easy completions and Frattini quotients, and isolate the distinction between topological density and abstract equality.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04Open item page →

A p-adic integer is encoded by a compatible sequence of residue digits

Example

For fixed digits $a_{0}, a_{1}, a_{2}, \dots \in {0, \dots, p - 1}$ , the sequence

$x_{n} : = a_{0} + a_{1} p + \dots + a_{n - 1} p^{n - 1} (m o d p^{n})$

defines an element of $Z_{p}$ .

Facts & Assumptions

Given: Digits $a_{i} \in {0, \dots, p - 1}$ for $i \geq 0$ .

[L1]

An element of $Z_{p}$ is a compatible residue-class tuple (The p-adic integers are the compatible residue-class tuples in the inverse limit of Z mod p^n).

Verification

technique · direct

1.1

For each $n \geq 1$ , reducing $x_{n + 1}$ modulo $p^{n}$ removes only the final term $a_{n} p^{n}$ , so $x_{n + 1} m o d p^{n} = x_{n}$ . Thus the tuple $(x_{n})_{n \geq 1}$ is compatible.

givenalgebra

2.1

By [L1], every compatible tuple defines an element of $Z_{p}$ . Hence the displayed digit data encodes a $p$ -adic integer.

L1step 1.1∎

ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04Open item page →

The partial sums of a p-adic expansion converge in the inverse-limit topology

Example

If $x \in Z_{p}$ is represented by digits $a_{0}, a_{1}, \dots$ , then the partial sums

$s_{n} : = a_{0} + a_{1} p + \dots + a_{n - 1} p^{n - 1} \in Z$

converge to $x$ in the inverse-limit topology on $Z_{p}$ .

Facts & Assumptions

Given: An element $x \in Z_{p}$ with compatible digit truncations $(s_{n} m o d p^{r})_{r}$ .

[L1]

The inverse-limit topology is the $p$ -adic metric topology (The inverse-limit topology on Zp agrees with the p-adic metric topology).

Verification

technique · direct

1.1

For each fixed $r \geq 1$ , once $n \geq r$ the residue of $s_{n}$ modulo $p^{r}$ is already $a_{0} + a_{1} p + \dots + a_{r - 1} p^{r - 1}$ , which is exactly the $r$ -th coordinate of $x$ . So the first $r$ coordinates of $s_{n}$ and $x$ agree for all sufficiently large $n$ .

givenalgebra

2.1

Step 1.1 says that for every $r$ , the sequence eventually lies in the basic neighbourhood of $x$ determined by the first $r$ coordinates. By [L1], that is exactly convergence to $x$ in the inverse-limit topology.

L1step 1.1∎

ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04Open item page →

A standard p-adic digit sequence is Cauchy and converges in Zp

Example

The sequence

$t_{n} : = 1 + p + \dots + p^{n - 1} \in Z \subseteq Z_{p}$

is Cauchy for the additive $p$ -adic metric and converges in $Z_{p}$ .

Facts & Assumptions

Given: The sequence $t_{n} = 1 + p + \dots + p^{n - 1}$ in $Z_{p}$ .

[F1]

The $p$ -adic metric measures how many initial residue coordinates agree (The p-adic metric on Zp is determined by the first coordinate at which two compatible residue systems differ).

[L1]

$Z_{p}$ is complete for that metric (Zp is Hausdorff, totally disconnected, and complete, and compact assuming Choice).

Verification

technique · direct

1.1

If $m > n$ , then $t_{m} - t_{n} = p^{n} (1 + p + \dots + p^{m - n - 1})$ , so the first $n$ residue coordinates of $t_{m}$ and $t_{n}$ agree. By [F1], this gives $d_{p} (t_{m}, t_{n}) \leq p^{- n}$ . Hence $(t_{n})$ is Cauchy.

F1givenalgebra

2.1

Since $Z_{p}$ is complete by [L1], the Cauchy sequence $(t_{n})$ converges to some element of $Z_{p}$ .

L1step 1.1∎

ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04Open item page →

A finite p-group is naturally isomorphic to its own pro-p completion

Example

If $P$ is a finite $p$ -group, then the canonical map $P \to {\hat{P}}^{(p)}$ is an isomorphism.

Facts & Assumptions

Given: A finite $p$ -group $P$ .

[F1]

The pro- $p$ completion is the inverse limit over all finite $p$ -group quotients (The pro-p completion of an abstract group is the inverse limit over its finite p-group quotients).

[L1]

A finite $p$ -group is itself a pro- $p$ group (A pro-p group is a profinite group that is an inverse limit of finite p-groups).

Verification

technique · direct

1.1

The subgroup $1 ⊴ P$ is one of the indexing subgroups in [F1], and its quotient is the initial object $P / 1 ≅ P$ of the quotient diagram: it has the natural quotient map to every $P / N$ .

F1givenalgebra

2.1

A compatible tuple is uniquely determined by its coordinate in the initial object $P / 1$ , and every element of $P / 1$ determines such a tuple by its images in the other quotients. Therefore ${\hat{P}}^{(p)} ≅ P$ , and the canonical completion map is exactly this isomorphism. The conclusion is consistent with [L1], since $P$ was already pro- $p$ to begin with.

L1step 1.1∎

ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-09-04Open item page →

The Frattini quotient of Zp is the one-dimensional vector space Fp

Example

For the additive pro- $p$ group $Z_{p}$ , one has

$Φ (Z_{p}) = p Z_{p} and Z_{p} / Φ (Z_{p}) ≅ F_{p} .$

Facts & Assumptions

Given: The additive pro- $p$ group $Z_{p}$ .

[L1]

For a finitely generated pro- $p$ group, the Frattini subgroup is $\overline{[G, G] G^{p}}$ (For a finitely generated pro-p group, the Frattini subgroup is the closure of [G,G]G^p).

Verification

technique · direct

1.1

The additive group $Z_{p}$ is abelian, so $[Z_{p}, Z_{p}] = 0$ . Its $p$ th-power subgroup in additive notation is $p Z_{p}$ , which is already closed. Therefore [L1] gives $Φ (Z_{p}) = p Z_{p}$ .

L1givenalgebra

2.1

Reduction modulo $p$ is a surjective homomorphism $Z_{p} \to Z / p Z$ with kernel $p Z_{p}$ . By step 1.1, the quotient by $Φ (Z_{p})$ is therefore $F_{p}$ .

step 1.1algebra∎

ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-09-04Open item page →

In Zp squared, topological generation is detected by the Frattini quotient coordinates

Example

In $G = Z_{p} \times Z_{p}$ , a pair $(u, v)$ topologically generates $G$ if and only if their reductions modulo $p$ form a basis of $F_{p}^{2}$ .

Facts & Assumptions

Given: The pro- $p$ group $G = Z_{p} \times Z_{p}$ .

[L1]

Topological generation in a finitely generated pro- $p$ group is detected on the Frattini quotient (A subset topologically generates a finitely generated pro-p group exactly when its image spans the Frattini quotient over Fp).

[F1]

The external direct product is coordinatewise (The external direct product $G \times H$ with componentwise multiplication).

Verification

technique · direct

1.1

By coordinatewise addition in [F1], the subgroup $p G$ consists of pairs whose two coordinates are divisible by $p$ . Therefore $G / p G ≅ (Z_{p} / p Z_{p}) \times (Z_{p} / p Z_{p}) ≅ F_{p}^{2}$ .

F1givenalgebra

2.1

The Frattini quotient of the abelian pro- $p$ group $G$ is exactly $G / p G$ , so [L1] says that a subset topologically generates $G$ exactly when its image spans $F_{p}^{2}$ . For a pair $(u, v)$ , spanning $F_{p}^{2}$ is the same as being a basis.

L1step 1.1algebra∎

CounterexampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04Open item page →

The embedded integers form an abstractly proper but topologically dense subgroup of Zp

Statement refuted

Every dense subgroup of $Z_{p}$ is all of $Z_{p}$ as an abstract subgroup.

Facts & Assumptions

Given: The embedded copy of $Z$ inside $Z_{p}$ .

[L1]

The image of $Z$ in $Z_{p}$ is dense (The canonical map from the integers into Zp is injective and has dense image).

[L2]

The additive group of $Z_{p}$ is not abstractly cyclic, even though the closure of $Z \cdot 1$ is all of $Z_{p}$ (The additive topological group of Zp is topologically generated by 1, although it is not abstractly cyclic).

Counterexample

technique · direct

1.1

By [L1], the embedded copy of $Z$ is dense in $Z_{p}$ .

L1given

2.1

The embedded copy of $Z$ is exactly $Z \cdot 1$ . By [L2], its closure is all of $Z_{p}$ but it is not equal to $Z_{p}$ as an abstract subgroup. Thus a dense subgroup can be proper.

L2step 1.1

3.1

Therefore the stated universal claim is false, and the embedded copy of $Z$ is the required counterexample.

step 1.1step 2.1∎

Pro P Groups and the P Adic Integers - Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

A p-adic integer is encoded by a compatible sequence of residue digits

Example

Facts & Assumptions

Verification

The partial sums of a p-adic expansion converge in the inverse-limit topology

Example

Facts & Assumptions

Verification

A standard p-adic digit sequence is Cauchy and converges in Zp

Example

Facts & Assumptions

Verification

A finite p-group is naturally isomorphic to its own pro-p completion

Example

Facts & Assumptions

Verification

The Frattini quotient of Zp is the one-dimensional vector space Fp

Example

Facts & Assumptions

Verification

In Zp squared, topological generation is detected by the Frattini quotient coordinates

Example

Facts & Assumptions

Verification

The embedded integers form an abstractly proper but topologically dense subgroup of Zp

Statement refuted

Facts & Assumptions

Counterexample

Sources