The Burnside problem asks whether a
finitely generated group
In algebra, a finitely generated group is a group ''G'' that has some finite generating set ''S'' so that every element of ''G'' can be written as the combination (under the group operation) of finitely many elements of ''S'' and of inverses of s ...
in which every element has finite
order must necessarily be a
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
. It was posed by
William Burnside
:''This English mathematician is sometimes confused with the Irish mathematician William S. Burnside (1839–1920).''
__NOTOC__
William Burnside (2 July 1852 – 21 August 1927) was an English mathematician. He is known mostly as an early rese ...
in 1902, making it one of the oldest questions in
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
and was influential in the development of
combinatorial group theory In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations. It is much used in geometric topology, the fundamental group of a simplicial complex having in a nat ...
. It is known to have a negative answer in general, as
Evgeny Golod
Evgenii Solomonovich Golod (russian: Евгений Соломонович Голод, 21 October 1935 – 5 July 2018) was a Russian mathematician who proved the Golod–Shafarevich theorem on class field tower
A tower is a tall structure ...
and
Igor Shafarevich
Igor Rostislavovich Shafarevich (russian: И́горь Ростисла́вович Шафаре́вич; 3 June 1923 – 19 February 2017) was a Soviet and Russian mathematician who contributed to algebraic number theory and algebraic geometry. ...
provided a counter-example in 1964. The problem has many refinements and variants (see
bounded and
restricted below) that differ in the additional conditions imposed on the orders of the group elements, some of which are still
open questions In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (i.e., no solution for it is kno ...
.
Brief history
Initial work pointed towards the affirmative answer. For example, if a group ''G'' is finitely generated and the order of each element of ''G'' is a divisor of 4, then ''G'' is finite. Moreover,
A. I. Kostrikin
Alexei Ivanovich Kostrikin (russian: Алексей Иванович Кострикин) (12 February 1929 – 22 September 2000) was a Russian mathematician, specializing in algebra and algebraic geometry.
Life
Kostrikin graduated from th ...
was able to prove in 1958 that among the finite groups with a given number of generators and a given prime exponent, there exists a largest one. This provides a solution for the
restricted Burnside problem for the case of prime exponent. (Later, in 1989,
Efim Zelmanov
Efim Isaakovich Zelmanov (russian: Ефи́м Исаа́кович Зе́льманов; born 7 September 1955 in Khabarovsk) is a Russian-American mathematician, known for his work on combinatorial problems in nonassociative algebra and group th ...
was able to solve the restricted Burnside problem for an arbitrary exponent.)
Issai Schur
Issai Schur (10 January 1875 – 10 January 1941) was a Russian mathematician who worked in Germany for most of his life. He studied at the University of Berlin. He obtained his doctorate in 1901, became lecturer in 1903 and, after a stay at the ...
had shown in 1911 that any finitely generated periodic group that was a subgroup of the group of invertible ''n'' × ''n'' complex matrices was finite; he used this theorem to prove the
Jordan–Schur theorem In mathematics, the Jordan–Schur theorem also known as Jordan's theorem on finite linear groups is a theorem in its original form due to Camille Jordan. In that form, it states that there is a function ''ƒ''(''n'') such that given a finite subgr ...
.
Nevertheless, the general answer to the Burnside problem turned out to be negative. In 1964, Golod and Shafarevich constructed an infinite group of Burnside type without assuming that all elements have uniformly bounded order. In 1968,
Pyotr Novikov
Pyotr Sergeyevich Novikov (russian: Пётр Серге́евич Но́виков; 15 August 1901, Moscow, Russian Empire – 9 January 1975, Moscow, Soviet Union) was a Soviet mathematician.
Novikov is known for his work on combinatorial proble ...
and
Sergei Adian
Sergei Ivanovich Adian, also Adyan ( hy, Սերգեյ Իվանովիչ Ադյան; russian: Серге́й Ива́нович Адя́н; 1 January 1931 – 5 May 2020), 4381, and hence for all multiples of those odd integers as well.
The solutio ...
supplied a negative solution to the bounded exponent problem for all odd exponents larger than 4381. In 1982,
A. Yu. Ol'shanskii found some striking counterexamples for sufficiently large odd exponents (greater than 10
10), and supplied a considerably simpler proof based on geometric ideas.
The case of even exponents turned out to be much harder to settle. In 1992, S. V. Ivanov announced the negative solution for sufficiently large even exponents divisible by a large power of 2 (detailed proofs were published in 1994 and occupied some 300 pages). Later joint work of Ol'shanskii and Ivanov established a negative solution to an analogue of Burnside problem for
hyperbolic group
In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a ''word hyperbolic group'' or ''Gromov hyperbolic group'', is a finitely generated group equipped with a word metric satisfying certain properties abstra ...
s, provided the exponent is sufficiently large. By contrast, when the exponent is small and different from 2, 3, 4 and 6, very little is known.
General Burnside problem
A group ''G'' is called
periodic if every element has finite order; in other words, for each ''g'' in ''G'', there exists some positive integer ''n'' such that ''g''
''n'' = 1. Clearly, every finite group is periodic. There exist easily defined groups such as the
''p''∞-group which are infinite periodic groups; but the latter group cannot be finitely generated.
General Burnside problem. If ''G'' is a finitely generated, periodic group, then is ''G'' necessarily finite?
This question was answered in the negative in 1964 by
Evgeny Golod
Evgenii Solomonovich Golod (russian: Евгений Соломонович Голод, 21 October 1935 – 5 July 2018) was a Russian mathematician who proved the Golod–Shafarevich theorem on class field tower
A tower is a tall structure ...
and
Igor Shafarevich
Igor Rostislavovich Shafarevich (russian: И́горь Ростисла́вович Шафаре́вич; 3 June 1923 – 19 February 2017) was a Soviet and Russian mathematician who contributed to algebraic number theory and algebraic geometry. ...
, who gave an example of an infinite
''p''-group that is finitely generated (see
Golod–Shafarevich theorem
In mathematics, the Golod–Shafarevich theorem was proved in 1964 by Evgeny Golod and Igor Shafarevich. It is a result in non-commutative homological algebra which solves the class field tower problem, by showing that class field towers can b ...
). However, the orders of the elements of this group are not ''a priori'' bounded by a single constant.
Bounded Burnside problem
![FreeBurnsideGroupExp3Gens2](https://upload.wikimedia.org/wikipedia/commons/c/c6/FreeBurnsideGroupExp3Gens2.png)
Part of the difficulty with the general Burnside problem is that the requirements of being finitely generated and periodic give very little information about the possible structure of a group. Therefore, we pose more requirements on ''G''. Consider a periodic group ''G'' with the additional property that there exists a least integer ''n'' such that for all ''g'' in ''G'', ''g''
''n'' = 1. A group with this property is said to be ''periodic with bounded exponent'' ''n'', or just a ''group with exponent'' ''n''. Burnside problem for groups with bounded exponent asks:
Burnside problem I. If ''G'' is a finitely generated group with exponent ''n'', is ''G'' necessarily finite?
It turns out that this problem can be restated as a question about the finiteness of groups in a particular family. The free Burnside group of rank ''m'' and exponent ''n'', denoted B(''m'', ''n''), is a group with ''m'' distinguished generators ''x''
1, ..., ''x
m'' in which the identity ''x
n'' = 1 holds for all elements ''x'', and which is the "largest" group satisfying these requirements. More precisely, the characteristic property of B(''m'', ''n'') is that, given any group ''G'' with ''m'' generators ''g''
1, ..., ''g
m'' and of exponent ''n'', there is a unique homomorphism from B(''m'', ''n'') to ''G'' that maps the ''i''th generator ''x
i'' of B(''m'', ''n'') into the ''i''th generator ''g
i'' of ''G''. In the language of
group presentations, free Burnside group B(''m'', ''n'') has ''m'' generators ''x''
1, ..., ''x
m'' and the relations ''x
n'' = 1 for each word ''x'' in ''x''
1, ..., ''x
m'', and any group ''G'' with ''m'' generators of exponent ''n'' is obtained from it by imposing additional relations. The existence of the free Burnside group and its uniqueness up to an isomorphism are established by standard techniques of group theory. Thus if ''G'' is any finitely generated group of exponent ''n'', then ''G'' is a
homomorphic image of B(''m'', ''n''), where ''m'' is the number of generators of ''G''. Burnside problem can now be restated as follows:
Burnside problem II. For which positive integers ''m'', ''n'' is the free Burnside group B(''m'', ''n'') finite?
The full solution to Burnside problem in this form is not known. Burnside considered some easy cases in his original paper:
*B(1, ''n'') is the
cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
of order ''n''.
*B(''m'', 2) is the
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
of ''m'' copies of the cyclic group of order 2 and hence finite.
[The key step is to observe that the identities ''a''2 = ''b''2 = (''ab'')2 = 1 together imply that ''ab'' = ''ba'', so that a free Burnside group of exponent two is necessarily abelian.]
The following additional results are known (Burnside, Sanov,
M. Hall):
*B(''m'', 3), B(''m'', 4), and B(''m'', 6) are finite for all ''m''.
The particular case of B(2, 5) remains open: it was not known whether this group is finite.
The breakthrough in solving the Burnside problem was achieved by
Pyotr Novikov
Pyotr Sergeyevich Novikov (russian: Пётр Серге́евич Но́виков; 15 August 1901, Moscow, Russian Empire – 9 January 1975, Moscow, Soviet Union) was a Soviet mathematician.
Novikov is known for his work on combinatorial proble ...
and
Sergei Adian
Sergei Ivanovich Adian, also Adyan ( hy, Սերգեյ Իվանովիչ Ադյան; russian: Серге́й Ива́нович Адя́н; 1 January 1931 – 5 May 2020), 4381, and hence for all multiples of those odd integers as well.
The solutio ...
in 1968. Using a complicated combinatorial argument, they demonstrated that for every
odd
Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric.
Odd may also refer to:
Acronym
* ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
number ''n'' with ''n'' > 4381, there exist infinite, finitely generated groups of exponent ''n''. Adian later improved the bound on the odd exponent to 665.
[ John Britton proposed a nearly 300 page alternative proof to the Burnside problem in 1973; however, Adian ultimately pointed out a flaw in that proof.] The latest improvement to the bound on odd exponent is 101 obtained by Adian himself in 2015. The case of even exponent turned out to be considerably more difficult. It was only in 1994 that Sergei Vasilievich Ivanov was able to prove an analogue of Novikov–Adian theorem: for any ''m'' > 1 and an even ''n'' ≥ 2
48, ''n'' divisible by 2
9, the group B(''m'', ''n'') is infinite; together with the Novikov–Adian theorem, this implies infiniteness for all ''m'' > 1 and ''n'' ≥ 2
48. This was improved in 1996 by I. G. Lysënok to ''m'' > 1 and ''n'' ≥ 8000. Novikov–Adian, Ivanov and Lysënok established considerably more precise results on the structure of the free Burnside groups. In the case of the odd exponent, all finite subgroups of the free Burnside groups were shown to be cyclic groups. In the even exponent case, each finite subgroup is contained in a product of two
dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ge ...
s, and there exist non-cyclic finite subgroups. Moreover, the
word
A word is a basic element of language that carries an semantics, objective or pragmatics, practical semantics, meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of w ...
and
conjugacy
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other ...
problems were shown to be effectively solvable in B(''m'', ''n'') both for the cases of odd and even exponents ''n''.
A famous class of counterexamples to the Burnside problem is formed by finitely generated non-cyclic infinite groups in which every nontrivial proper subgroup is a finite
cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
, the so-called
Tarski Monsters. First examples of such groups were constructed by
A. Yu. Ol'shanskii in 1979 using geometric methods, thus affirmatively solving O. Yu. Schmidt's problem. In 1982 Ol'shanskii was able to strengthen his results to establish existence, for any sufficiently large
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
''p'' (one can take ''p'' > 10
75) of a finitely generated infinite group in which every nontrivial proper subgroup is a
cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
of order ''p''. In a paper published in 1996, Ivanov and Ol'shanskii solved an analogue of the Burnside problem in an arbitrary
hyperbolic group
In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a ''word hyperbolic group'' or ''Gromov hyperbolic group'', is a finitely generated group equipped with a word metric satisfying certain properties abstra ...
for sufficiently large exponents.
Restricted Burnside problem
Formulated in the 1930s, it asks another, related, question:
Restricted Burnside problem. If it is known that a group ''G'' with ''m'' generators and exponent ''n'' is finite, can one conclude that the order of ''G'' is bounded by some constant depending only on ''m'' and ''n''? Equivalently, are there only finitely many ''finite'' groups with ''m'' generators of exponent ''n'', up to isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
?
This variant of the Burnside problem can also be stated in terms of certain universal groups with ''m'' generators and exponent ''n''. By basic results of group theory, the intersection of two subgroups of finite
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
in any group is itself a subgroup of finite index. Let ''M'' be the intersection of all subgroups of the free Burnside group B(''m'', ''n'') which have finite index, then ''M'' is a
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
of B(''m'', ''n'') (otherwise, there exists a subgroup ''g''
−1''Mg'' with finite index containing elements not in ''M''). One can therefore define a group B
0(''m'', ''n'') to be the factor group B(''m'', ''n'')/''M''. Every finite group of exponent ''n'' with ''m'' generators is a homomorphic image of B
0(''m'', ''n'').
The restricted Burnside problem then asks whether B
0(''m'', ''n'') is a finite group.
In the case of the prime exponent ''p'', this problem was extensively studied by
A. I. Kostrikin
Alexei Ivanovich Kostrikin (russian: Алексей Иванович Кострикин) (12 February 1929 – 22 September 2000) was a Russian mathematician, specializing in algebra and algebraic geometry.
Life
Kostrikin graduated from th ...
during the 1950s, prior to the negative solution of the general Burnside problem. His solution, establishing the finiteness of B
0(''m'', ''p''), used a relation with deep questions about identities in
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
s in finite characteristic. The case of arbitrary exponent has been completely settled in the affirmative by
Efim Zelmanov
Efim Isaakovich Zelmanov (russian: Ефи́м Исаа́кович Зе́льманов; born 7 September 1955 in Khabarovsk) is a Russian-American mathematician, known for his work on combinatorial problems in nonassociative algebra and group th ...
, who was awarded the
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award ho ...
in 1994 for his work.
Notes
References
Bibliography
*
S. I. Adian (1979) ''The Burnside problem and identities in groups''. Translated from the Russian by John Lennox and James Wiegold.
Ergebnisse der Mathematik und ihrer Grenzgebiete
''Ergebnisse der Mathematik und ihrer Grenzgebiete''/''A Series of Modern Surveys in Mathematics'' is a series of scholarly monographs published by Springer Science+Business Media. The title literally means "Results in mathematics and related area ...
esults in Mathematics and Related Areas 95. Springer-Verlag, Berlin-New York. .
* Translation in
*
*
* A. I. Kostrikin (1990) ''Around Burnside''. Translated from the Russian and with a preface by
James Wiegold. ''Ergebnisse der Mathematik und ihrer Grenzgebiete'' (3)
esults in Mathematics and Related Areas (3) 20. Springer-Verlag, Berlin. .
* Translation in
* A. Yu. Ol'shanskii (1989) ''Geometry of defining relations in groups''. Translated from the 1989 Russian original by Yu. A. Bakhturin (1991) ''Mathematics and its Applications'' (Soviet Series), 70. Dordrecht: Kluwer Academic Publishers Group. .
* Translation in
* Translation in {{cite journal , journal= Mathematics of the USSR-Sbornik, volume=72 , year=1992 , issue=2 , pages=543–565 , title=A Solution of the Restricted Burnside Problem for 2-groups , last1=Zel'manov , first1=E I, doi=10.1070/SM1992v072n02ABEH001272, bibcode=1992SbMat..72..543Z
External links
History of the Burnside problemat
MacTutor History of Mathematics archive
The MacTutor History of Mathematics archive is a website maintained by John J. O'Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland. It contains detailed biographies on many historical and contemporary mathemati ...
Group theory
Unsolved problems in mathematics