# P2860: A Tarski monster of exponent five

- ID: `P2860`
- Reference: `tarski-monster-exponent-five`
- Page: https://theoremdb.org/statements/P2860
- Record maturity: Reviewed problem with recorded work

## Problem

Does there exist an infinite group \(G\) of exponent \(5\) such that every nontrivial proper subgroup of \(G\) is cyclic of order \(5\)?

### Problem setup

- **Remark.** A group has exponent \(5\) if \(g^5=1\) for every \(g\in G\).
- **Definition.** A Tarski monster is an infinite group whose nontrivial proper subgroups all have the same prime order.
- **Definition.** Cyclic of order \(5\) means generated by one nonidentity element and containing exactly five elements.

### What counts as a solution

- For a positive answer, construct a group \(G\) and prove infinitude, exponent five, and the assertion that every nontrivial proper subgroup is cyclic of order five.
- For a negative answer, prove that no infinite exponent-five group can have the stated subgroup structure.

## Status

Unresolved in this packet after the dated source check. Strongest checked result: The MathOverflow page has no answers and its comments connect the case to the small odd-exponent Burnside problem. The checked modern theorem proves infinitude for much larger odd exponents and does not reach exponent five. Exact unresolved remainder: For a positive answer, construct a group \(G\) and prove infinitude, exponent five, and the assertion that every nontrivial proper subgroup is cyclic of order five. For a negative answer, prove that no infinite exponent-five group can have the stated subgroup structure. [2](#reference-2) [3](#reference-3) [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (Dated status and exact unresolved remainder).** Unresolved in this packet after the dated source check. Strongest checked result: The MathOverflow page has no answers and its comments connect the case to the small odd-exponent Burnside problem. The checked modern theorem proves infinitude for much larger odd exponents and does not reach exponent five. Exact unresolved remainder: For a positive answer, construct a group \(G\) and prove infinitude, exponent five, and the assertion that every nontrivial proper subgroup is cyclic of order five. For a negative answer, prove that no infinite exponent-five group can have the stated subgroup structure.

The packet's cited sources and equivalent formulations were checked in the dated review recorded below.

Strongest checked result: The MathOverflow page has no answers and its comments connect the case to the small odd-exponent Burnside problem. The checked modern theorem proves infinitude for much larger odd exponents and does not reach exponent five.

Exact unresolved remainder: For a positive answer, construct a group \(G\) and prove infinitude, exponent five, and the assertion that every nontrivial proper subgroup is cyclic of order five. For a negative answer, prove that no infinite exponent-five group can have the stated subgroup structure.

### Background and intake notes

The target isolates the smallest prime absent from known large-exponent constructions. Quotient presentations and machine-checked finite consequences can document progress, though proving infinitude and subgroup rigidity requires a global argument.

- Original intake status: UNKNOWN as of 2026-07-27. The MathOverflow page has no answers and its comments connect the case to the small odd-exponent Burnside problem. The checked modern theorem proves infinitude for much larger odd exponents and does not reach exponent five.
- On 2026-07-27 the unanswered MathOverflow question and all comments were checked. No construction or impossibility proof is posted.
- Olshanskii's Tarski-monster constructions work for sufficiently large primes. The exact exponent five lies far below the range supplied by those methods.
- Atabekyan and Ivanov, arXiv:2303.15997, prove infinitude of free Burnside groups for odd exponents at least 557. Their quantitative threshold leaves exponent five untouched.
- The restricted Burnside group of exponent five is finite, while the ordinary free Burnside group \(B(2,5)\) has long-standing unresolved behavior in the sources checked. A Tarski monster would in particular provide an infinite exponent-five group.
- Trap: an infinite group generated by elements of order five need not have exponent five, and an infinite exponent-five group need not have the Tarski subgroup property.

- Recorded example: Every nonidentity element in a putative group generates a subgroup of order five. The defining condition then says any two elements that generate a proper subgroup must lie in one such cyclic subgroup.

### Open directions

- **Route 1** (reported): For a positive answer, construct a group \(G\) and prove infinitude, exponent five, and the assertion that every nontrivial proper subgroup is cyclic of order five. For a negative answer, prove that no infinite exponent-five group can have the stated subgroup structure. [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `tarski-monster-exponent-five`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>MathOverflow question 138368, “A Tarski monster of exponent five,” checked 2026-08-01. Original CC0 existence statement written after checking the unanswered question, its comments, and current bounds in the odd-exponent Burnside literature. https://mathoverflow.net/questions/138368/tarski-monster-group-with-prime-5
   - Also cited at Editorial research route recorded 2026-08-01.
   - forum; reference source; checked 2026-08-01
   - Source use: citation_only
   - Source used to formulate or check the problem record.
   - Source used to assess the problem's recorded status.
   - For A Tarski monster of exponent five, the reviewed source scope is Original CC0 existence statement written after checking the unanswered question, its comments, and current bounds in the odd-exponent Burnside literature.. The packet makes no inference beyond that cited scope.
   - Source named by the research packet.
2. <a id="reference-2"></a>Agatha Atkarskaya, Eliyahu Rips, and Katrin Tent, “The Burnside problem for odd exponents,” arXiv:2303.15997 (2023). abstract and main theorem proving B(m,n) infinite for m >= 2 and odd n >= 557 https://arxiv.org/abs/2303.15997
   - preprint; reference source; arXiv:2303.15997, checked 2026-08-01; checked 2026-08-01
   - Source use: citation_only
   - Source used to assess the problem's recorded status.
   - For A Tarski monster of exponent five, this source gives the large-odd-exponent boundary without reaching exponent five or constructing a Tarski monster there.
3. <a id="reference-3"></a>George Havas, G. E. Wall, and J. W. Wamsley, “The two generator restricted Burnside group of exponent five,” Bulletin of the Australian Mathematical Society 10(3) (1974), 459-470. DOI 10.1017/S0004972700041137. abstract and computation of the two-generator restricted Burnside group of exponent five, of order 5^34 and class 12 https://doi.org/10.1017/S0004972700041137
   - scholarly_publication; reference source; checked 2026-08-01
   - Source use: citation_only
   - Source used to assess the problem's recorded status.
   - For A Tarski monster of exponent five, this source supplies exact exponent-five finite-quotient information; the restricted Burnside group is not the infinite group needed for a Tarski monster.