[#P2860] A Tarski monster of exponent five
Problem. Does there exist an infinite group \(G\) of exponent \(5\) such that every nontrivial proper subgroup of \(G\) is cyclic of order \(5\)?
1Context
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.
2Problem setup
Remark 1. A group has exponent \(5\) if \(g^5=1\) for every \(g\in G\).
Definition 1. A Tarski monster is an infinite group whose nontrivial proper subgroups all have the same prime order.
Definition 2. Cyclic of order \(5\) means generated by one nonidentity element and containing exactly five elements.
3What 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.
1Status
Current status (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.[2][3][1]
1Records
Notes and companion material
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.
- Independent source, duplicate, exact-title, and equivalent-formulation review completed on 2026-08-01.
Recorded example 1. 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.
2See also
How to cite
TheoremDB contributors, “A Tarski monster of exponent five,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/tarski-monster-exponent-fiveThis page as plain text: tarski-monster-exponent-five.md
This problem includes 2 records joined by 2 typed links, sourced from mathoverflow.net[1], current as of August 1, 2026.
1References
- Packet source. MathOverflow question 138368, “A Tarski monster of exponent five,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗forum · reference source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at Original CC0 existence statement written after checking the unanswered question, its comments, and current bounds in the odd-exponent Burnside literature.Also cited at Editorial research route recorded 2026-08-01.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.
- arXiv preprint 2303.15997, linked primary source for “A Tarski monster of exponent five,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗preprint · primary source · arXiv:2303.15997, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at abstract and main theorem proving B(m,n) infinite for m >= 2 and odd n >= 557.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.
- 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. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at abstract and computation of the two-generator restricted Burnside group of exponent five, of order 5^34 and class 12.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.
Original CC0 textbook restatement.