Open problem, Algebra, posed 1902

Burnside problem

Partially solved

If $G$ is a finitely generated group in which every element has finite order (the General Burnside Problem), or in which every element $g \in G$ satisfies $g^n = 1$ for a fixed integer $n \ge 1$ (the Bounded Burnside Problem for the free Burnside group $B(m, n)$ ), must $G$ be finite? In the Restricted Burnside Problem, is there a largest finite $m$ -generated group of exponent $n$ ?

Research frontier as of 2026

As of 2026, while the General Burnside Problem (false) and the Restricted Burnside Problem (true) are completely resolved, the Bounded Burnside Problem remains open for intermediate exponents. Specifically, $B(m, n)$ is known to be finite for $m \ge 2$ only when $n \in \{1, 2, 3, 4, 6\}$ , and it is a famous open question whether the $2$ -generator group $B(2, 5)$ of exponent $5$ (or $B(2, 8)$ of exponent $8$ ) is finite or infinite.

Best known results

The Restricted Burnside Problem holds for all $m$ and $n$ : every $m$ -generated finite group of exponent $n$ has order bounded by a function $f(m, n)$ (Kostrikin 1958 for prime $n$ ; Zelmanov 1990–1991 for prime-power $n$ ; Hall–Higman 1956 reduction for general $n$ ). For $B(2, 5)$ , the largest finite quotient $R(2, 5)$ has order $5^{34}$ (Havas, Newman, and Vaughan-Lee 1990).
The free Burnside group $B(m, n)$ ( $m \ge 2$ ) is finite for $n \in \{1, 2, 3, 4, 6\}$ and infinite for all odd $n \ge 665$ (Novikov–Adian 1968, Adian 1975) and for large even exponents (Ivanov 1994, Lysenok 1996).

Tools and where they stop

Tool	Achieved	Where it stops
Graded Lie algebras and Jordan algebra identities	Bounds the nilpotency class of finite $p$ -groups of exponent $p^k$ by translating group commutators into Engel-like identities in Lie algebras, completely solving the Restricted Burnside Problem	Applies only to residual finiteness (finite quotients) and cannot rule out the existence of an infinite simple or non-residually-finite quotient inside $B(2, 5)$
Novikov–Adian inductive word analysis and geometric small cancellation (van Kampen diagrams)	Controls periodic words across inductive ranks to prove infinitude of $B(m, n)$ for large odd and even exponents	Small-cancellation curvature estimates break down when the exponent $n$ is small (such as $n = 5, 7, 8$ ), where overlapping periodic relations interact too tightly

Open questions

Is the free Burnside group $B(2, 5)$ of rank $2$ and exponent $5$ finite (of order $5^{34}$ ) or infinite?
What is the smallest exponent $n$ for which $B(2, n)$ is infinite?

References

Sergei I. Adian (1979). The Burnside Problem and Identities in Groups
Efim I. Zelmanov (1990). Solution of the restricted Burnside problem for groups of odd exponent · DOI:10.1070/IM1991v036n01ABEH001946
Efim I. Zelmanov (1991). Solution of the restricted Burnside problem for 2-groups · DOI:10.1070/SM1992v072n02ABEH001272
Sergei V. Ivanov (1994). The free Burnside groups of sufficiently large exponents · DOI:10.1142/S0218196794000026