A history of the Burnside problem


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

  2. A Group GG is said to be periodic of bounded exponent if there exists nNn \in \mathbb{N} with gn=1g^{n} = 1 for all gGg \in G. The minimal such nn is called the exponent of GG.
It is clear that any finite group is periodic. In his 1902 paper, Burnside [1] 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 FmF_{m} denote the free group of rank mm. For a fixed nn let FmnF_{m}^{n} denote the subgroup of FmF_{m} generated by gng^{n} for gGg \in G.
Then FmnF_{m}^{n} is a normal subgroup of FmF_{m} (it is even an invariant subgroup), and we define the Burnside Group B(m,n)B(m, n) to be the factor group Fm/FmnF_{m}/ F_{m}^{n}.

Burnside showed a number of results in his 1902 paper;
  1. B(1,n)CnB(1, n) \cong C_{n}
  2. B(m,2)B(m, 2) is an elementary abelian group of order 2n2^{n} (a direct product of nn copies of C2C_{2})
  3. B(m,3)B(m, 3) is finite of order ≤ 32m13^{2m-1}
  4. B(2,4)B(2, 4) is finite of order ≤ 2122^{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 [2])
A finitely generated linear group which is finite dimensional and has finite exponent is finite i.e. any subgroup of GL(n,C)GL(n,\mathbb{C}) with bounded exponent is finite.

Theorem (Schur, 1911 [3])
Every finitely generated periodic subgroup of GL(n,C)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,nm, n then we may factor B(m,n)B(m, n) by the intersection of all subgroups of finite index to obtain B0(m,n)B_{0}(m,n), the universal finite mm-generator group of exponent nn having all other finite mm-generator groups of exponent nn as homomorphic images.

Note that if B(m,n)B(m,n) is finite then B0(m,n)B(m,n)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 [6], appeared specifically addressing the RBP, and not until 1950 that the term "Restricted Burnside Problem" was coined by Magnus [7].

1933
Levi, Van der Waerden [4] (independently) showed that B(m,3)B(m, 3) has order 3c,c=m+mC2+mC33^{c}, c = m + _{m}C_{2} + _{m}C_{3} and is a metabelian group of nilpotency class 3.

1940
Sanov [5] proved that B(m,4)B(m, 4) is finite.

1954
Tobin [8] showed that B(2,4)B(2, 4) has order 2122^{12}, and gave a presentation.

1955
Kostrikin[9] established that B0(2,5)B_{0}(2, 5) exists.

1956
Higman[10] proved that B0(m,5)B_{0}(m, 5) exists.

P Hall and G Higman [11] showed that B0(m,6)B_{0}(m, 6) exists and has order 2a3b2^{a}3^{b} where a=1+(m1)3c,b=1+(m1)2m,c=m+mC2+mC3a = 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. [12] proved that B(m,6)B(m, 6) is finite, a contribution which was described as a "heroic piece of calculation" by one reviewer.

Kostrikin[13] showed that B0(m,p)B_{0}(m, p) exists for all pp prime.

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

Theorem (Hall-Higman, 1956 [11])
Suppose that n=p1k1....,prkrn = p_{1}^{k_{1}}. ... , p_{r}^{k_{r}} with p1,...,prp_{1}, ... , p_{r} distinct primes.
Assume

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

(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 nn 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 B0(m,n)B_{0}(m,n) exists for all m,nm, n we need only (!) show that B0(m,pk)B_{0}(m, p^{k}) exists for all mm and prime powers pkp^{k}. Kostrikin had "shown" that B0(m,p)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)B(m, n) is infinite for nn odd, n>71n > 71. Novikov published a collection of ideas and theorems [14], 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 [15] provided a counter-example to the General Burnside Problem -- an infinite, finitely generated, periodic group.

1968
S I Adian, P S Novikov [16] proved that B(m,n)B(m, n) is infinite for nn odd, n4381n ≥ 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 [17] proved that B(m,n)B(m, n) is infinite if nn odd, n665n ≥ 665, improving the Adian-Novikov result of 1968.

1982
Ol'shanskii showed that given pp a prime, p>1075p > 10^{75}, then there is an infinite group, every proper subgroup of which is cyclic of order pp. (This is called the Tarski Monster)

1994
S V Ivanov published his proof that B(m,n)B(m, n) is infinite for m2m ≥ 2 and n248n ≥ 2^{48}.

1996
I G Lysenok proved that B(m,n)B(m, n) is infinite for m2m ≥ 2 and n8000n ≥ 8000.
It is still an open question whether B(2,5)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).]

Additional Resources (show)

Other websites about Burnside problem:

  1. Mathworld
  2. S V Ivanov (pdf)

Written by J J O'Connor and E F Robertson. Thanks to Andrew Isherwood.
Last Update July 2002