# A history of the Burnside problem

Some definitions
1. A Group $G$ is said to be periodic if for all $g \in G$ there exists $n \in \mathbb{N}$ with $g^{n} = 1$.
(Note that the number $n$ may depend on the element $g$.)

2. A Group $G$ is said to be periodic of bounded exponent if there exists $n \in \mathbb{N}$ with $g^{n} = 1$ for all $g \in G$. The minimal such $n$ is called the exponent of $G$.
It is clear that any finite group is periodic. In his 1902 paper, Burnside  introduced what he termed "a still undetermined point" in the theory of groups:

General Burnside Problem:
Is a finitely generated periodic group necessarily finite?

Burnside immediately suggested the "easier" question:

Burnside Problem:
Is a finitely generated periodic group of bounded exponent necessarily finite?

Definition
Let $F_{m}$ denote the free group of rank $m$. For a fixed $n$ let $F_{m}^{n}$ denote the subgroup of $F_{m}$ generated by $g^{n}$ for $g \in G$.
Then $F_{m}^{n}$ is a normal subgroup of $F_{m}$ (it is even an invariant subgroup), and we define the Burnside Group $B(m, n)$ to be the factor group $F_{m}/ F_{m}^{n}$.

Burnside showed a number of results in his 1902 paper;
1. $B(1, n) \cong C_{n}$
2. $B(m, 2)$ is an elementary abelian group of order $2^{n}$ (a direct product of $n$ copies of $C_{2}$)
3. $B(m, 3)$ is finite of order ≤ $3^{2m-1}$
4. $B(2, 4)$ is finite of order ≤ $2^{12}$. (in fact Burnside claimed equality)
Burnside and Schur made early progress on the problems in two papers, which confirmed that the problem would certainly not be straightforward:

Theorem (Burnside, 1905 )
A finitely generated linear group which is finite dimensional and has finite exponent is finite i.e. any subgroup of $GL(n,\mathbb{C})$ with bounded exponent is finite.

Theorem (Schur, 1911 )
Every finitely generated periodic subgroup of $GL(n,\mathbb{C})$ is finite.

These results imply that any counterexample to the Burnside Problems will have to be difficult, i.e. not expressible in terms of the well-known linear groups. After this initial flurry of results, no more progress was made on the Problems until the early 1930's, when the topic was resurrected by the suggestion of a variant on the original problem:

Restricted Burnside Problem:
Are there only finitely many finite m-generator groups of exponent n?

If the Restricted Burnside Problem has a positive solution for some $m, n$ then we may factor $B(m, n)$ by the intersection of all subgroups of finite index to obtain $B_{0}(m,n)$, the universal finite $m$-generator group of exponent $n$ having all other finite $m$-generator groups of exponent $n$ as homomorphic images.

Note that if $B(m,n)$ is finite then $B_{0}(m,n) \cong B(m,n)$.

Despite this formulation having been present on the seminar circuit in the 1930's, it was not until 1940 that the first paper, by Grün , appeared specifically addressing the RBP, and not until 1950 that the term "Restricted Burnside Problem" was coined by Magnus .

1933
Levi, Van der Waerden  (independently) showed that $B(m, 3)$ has order $3^{c}, c = m + _{m}C_{2} + _{m}C_{3}$ and is a metabelian group of nilpotency class 3.

1940
Sanov  proved that $B(m, 4)$ is finite.

1954
Tobin  showed that $B(2, 4)$ has order $2^{12}$, and gave a presentation.

1955
Kostrikin established that $B_{0}(2, 5)$ exists.

1956
Higman proved that $B_{0}(m, 5)$ exists.

P Hall and G Higman  showed that $B_{0}(m, 6)$ exists and has order $2^{a}3^{b}$ where $a = 1 + (m - 1)3^{c} , b = 1 + (m - 1)2^{m} , c = m + _{m}C_{2} + _{m}C_{3}$ and is hence soluble of derived length 3.

1958
Marshall Hall Jr.  proved that $B(m, 6)$ is finite, a contribution which was described as a "heroic piece of calculation" by one reviewer.

Kostrikin showed that $B_{0}(m, p)$ exists for all $p$ prime.

The 1956 Hall-Higman paper contains a remarkable reduction theorem for the Restricted Burnside Problem:

Theorem (Hall-Higman, 1956 )
Suppose that $n = p_{1}^{k_{1}}. ... , p_{r}^{k_{r}}$ with $p_{1}, ... , p_{r}$ distinct primes.
Assume

1. The RBP holds for groups of exponent $p_{i}^{k_{i}}$,
2. There are finitely many finite simple groups of exponent $n$,
3. The outer automorphism group $Out(G) = Aut(G)/Inn(G)$ is soluble for any finite simple group of exponent $n$.
Then the RBP holds for groups of exponent $n$.

(Note that iii. above is the so-called Schreier Conjecture)

Now (moving ahead), the classification of finite simple groups in the 1980's shows that ii. and iii. hold. Even earlier it was known for $n$ odd by Feit-Thompson (the "odd-order paper" of 1962), and at the time of publication must have been a reasonable conjecture.

Consequently, to prove that $B_{0}(m,n)$ exists for all $m, n$ we need only (!) show that $B_{0}(m, p^{k})$ exists for all $m$ and prime powers $p^{k}$. Kostrikin had "shown" that $B_{0}(m, p)$ exists.

In 1989 Zelmanov announced his proof of a positive solution of the Restricted Burnside Problem and was awarded a Fields medal for this in 1994.

1959
Turning back to the original Burnside Problems, Novikov announced that $B(m, n)$ is infinite for $n$ odd, $n > 71$. Novikov published a collection of ideas and theorems , but no definitive proof was forthcoming. John Britton suspected Novikov's proof was wrong and he began to work on the problem.

1964
Golod and Shafarevich  provided a counter-example to the General Burnside Problem -- an infinite, finitely generated, periodic group.

1968
S I Adian, P S Novikov  proved that $B(m, n)$ is infinite for $n$ odd, $n ≥ 4381$ with an epic combinatorial proof based upon Novikov's earlier efforts.

This saddened Britton since he was close to publishing himself, but he continued and finished in 1970. His paper was published in 1973, but Adian discovered that it was wrong. There was not a single error in any lemma. However in order to apply them simultaneously the inequalities needed to make their hypotheses valid were inconsistent. Britton never really recovered, and this was to be the last major research paper he published.

1975
S I Adian  proved that $B(m, n)$ is infinite if $n$ odd, $n ≥ 665$, improving the Adian-Novikov result of 1968.

1982
Ol'shanskii showed that given $p$ a prime, $p > 10^{75}$, then there is an infinite group, every proper subgroup of which is cyclic of order $p$. (This is called the Tarski Monster)

1994
S V Ivanov published his proof that $B(m, n)$ is infinite for $m ≥ 2$ and $n ≥ 2^{48}$.

1996
I G Lysenok proved that $B(m, n)$ is infinite for $m ≥ 2$ and $n ≥ 8000$.
It is still an open question whether $B(2, 5)$ is finite or not.

### References (show)

1. W Burnside, On an unsettled question in the theory of discontinuous groups, Quart.J.Math. 33 (1902), 230-238.
2. W Burnside, On criteria for the finiteness of the order of a group of linear substitutions, Proc.London Math. Soc. (2) 3 (1905), 435-440.
3. I Schur, Über Gruppen periodischer substitutionen, Sitzungsber. Preuss. Akad. Wiss. (1911), 619-627.
4. F Levi / B L Van der Waerden, Über eine besonderen Klasse von Gruppen, Abh. Math. Sem. Hamburg. Univ. 9 (1933), 154-156 / Math. Zeit 32 (1930), 315-318.
5. I N Sanov, Solution of Burnside's problem for n = 4, Leningrad State University Annals (Uchenyi Zapiski) Math. Ser. 10 (1940),166-170 (Russian).
6. O Grün, Zusammenhang zwischen Potenzbildung und Kommutatorbildung, J.f.d. reine u. angew.Math. 182 (1940), 158-177.
7. W Magnus, A connection between the Baker-Hausdorff formula and a problem of Burnside, Ann of Math. 52 (1950), 111-126; Errata, Ann. Of Math. 57 (1953), 606.
8. J J Tobin, On groups with exponent 4, Thesis , University of Manchester (1954).
9. A I Kostrikin, Lösung des abgeschwächten Burnsideschen Problems für den Exponenten 5. Izv. Akad. Nauk SSSR, Ser. Mat. 19, No. 3 (1955), 233-244 .
10. G Higman, On finite groups of exponent five, Proc. Cambridge Philos. Soc. 52 (1956), 381-390.
11. P Hall, P and G Higman, On the p-length of p-soluble groups and reduction theorems for Burnside's Problem, Proc. London Math. Soc. (3) 6 (1956), 1-42
12. M Hall Jr., Solution of the Burnside Problem for Exponent Six, Illinois J. of Math. 2 (1958), 764-786.
13. A I Kotsrikin, The Burnside Problem, Izv. Akad. Nauk. SSSR. Ser. Math. 23, (1958) 3-34 (Russian). American Math. Soc. Translations (2) 36 (1964), 63-99.
14. P S Novikov, On periodic groups, Dokl. Akad. Nauk SSSR Ser. Mat. 27 (1959), 749-752.
15. E S Golod, On nil algebras and finitely residual groups, Izv. Akad. Nauk SSSR. Ser. Mat. 1975, (1964), 273-276.
16. S I Adjan and P S Novikov, On infinite periodic groups I, II, III, Izv. Akad. Nauk SSSR. Ser. Mat. 32 (1968), 212-244; 251-524; 709-731.
17. S I Adjan, The Burnside problems and identities in groups, (Moscow, 1975). [Translated from the Russian by J Lennox and J Wiegold (Berlin, 1979).]