Some definitions

1. A Group G is said to be periodic if for all g ∈ G there exists n ∈ N with gⁿ = 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 ∈ N with gⁿ = 1 for all g ∈ 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 [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 F_m denote the free group of rank m. For a fixed n let F_mⁿ denote the subgroup of F_m generated by gⁿ for g ∈ G. 
Then F_mⁿ 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_mⁿ .

Burnside showed a number of results in his 1902 paper;

1. B(1, n)  C_n
2. B(m, 2) is an elementary abelian group of order 2ⁿ (a direct product of n copies of C₂)
3. B(m, 3) is finite of order ≤ 3^2m-1
4. B(2, 4) is finite of order ≤ 2¹². (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) with bounded exponent is finite.

Theorem (Schur, 1911 [3]) 
Every finitely generated periodic subgroup of GL(n,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 B₀(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 B₀(m,n) = 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].

It is still an open question whether B(2, 5) is finite or not.

W Burnside, On an unsettled question in the theory of discontinuous groups, Quart.J.Math. 33 (1902), 230-238.

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1. 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.
2. I Schur, Über Gruppen periodischer substitutionen, Sitzungsber. Preuss. Akad. Wiss. (1911), 619-627.
3. 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.
4. I N Sanov, Solution of Burnside's problem for n = 4, Leningrad State University Annals (Uchenyi Zapiski) Math. Ser. 10 (1940),166-170 (Russian).
5. O Grün, Zusammenhang zwischen Potenzbildung und Kommutatorbildung, J.f.d. reine u. angew.Math. 182 (1940), 158-177.
6. 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.
7. J J Tobin, On groups with exponent 4, Thesis , University of Manchester (1954).
8. A I Kostrikin, Lösung des abgeschwächten Burnsideschen Problems für den Exponenten 5. Izv. Akad. Nauk SSSR, Ser. Mat. 19, No. 3 (1955), 233-244 .
9. G Higman, On finite groups of exponent five, Proc. Cambridge Philos. Soc. 52 (1956), 381-390.
10. 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
11. M Hall Jr., Solution of the Burnside Problem for Exponent Six, Illinois J. of Math. 2 (1958), 764-786.
12. 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.
13. P S Novikov, On periodic groups, Dokl. Akad. Nauk SSSR Ser. Mat. 27 (1959), 749-752.
14. E S Golod, On nil algebras and finitely residual groups, Izv. Akad. Nauk SSSR. Ser. Mat. 1975, (1964), 273-276.
15. 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.
16. S I Adjan, The Burnside problems and identities in groups, (Moscow, 1975). [Translated from the Russian by J Lennox and J Wiegold (Berlin, 1979).]

مواضيع ذات صلة

word problems for groups

The fundamental theorem of algebra

The development of Ring Theory

The development of group theory

Quadratic, cubic and quartic equations

Matrices and determinants

Abstract linear spaces

A history of set theory