Burnside's problem
Encyclopedia
The Burnside problem, posed by William Burnside
in 1902 and one of the oldest and most influential questions in group theory
, asks whether a finitely generated
group
in which every element has finite order
must necessarily be a finite group
. In plain language, if by looking at individual elements of a group we suspect that the whole group is finite, must it indeed be true? The problem has many variants (see bounded and restricted below) that differ in the additional conditions imposed on the orders of the group elements.
(in general) proved that, among the finite groups with given number of generators and exponent, there exists a largest one. Issai Schur
showed that any finitely generated periodic group that was a subgroup of the group of invertible n x n complex matrices was finite; he used this theorem to prove the Jordan–Schur theorem
.
Nevertheless, the general answer to Burnside's 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 and Sergei Adian
's 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 1010), 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's problem for hyperbolic group
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.
if every element has finite order; in other words, for each g in G, there exists some positive integer n such that gn = 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.
The general Burnside problem can be posed as follows:
This question was answered in the negative in 1964 by Evgeny Golod
and Igor Shafarevich
, who gave an example of an infinite p-group
that is finitely generated (see Golod-Shafarevich theorem). However, the orders of the elements of this group are not a priori bounded by a single constant.
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 x1,…,xm in which the identity xn = 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 g1,…,gm and of exponent n, there is a unique homomorphism from B(m, n) to G that maps the ith generator xi of B(m, n) into the ith generator gi of G. In the language of group presentations
, free Burnside group B(m, n) has m generators x1,…,xm and the relations xn = 1 for each word x in x1,…,xm, 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 a homomorphic image
of B(m, n), where m is the number of generators of G. Burnside's problem can now be restated as follows:
The full solution to Burnside's problem in this form is not known. Burnside considered some easy cases in his original paper:
The following additional results are known (Burnside, Sanov, M. Hall
):
The particular case of B(2, 5) remains open: it was not known whether this group is finite.
The breakthrough in Burnside's problem was achieved by Pyotr Novikov and Sergei Adian
in 1968.
Using a complicated combinatorial argument, they demonstrated that for every odd
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. The case of even exponent turned out to be considerably more difficult. It was only in 1992 that Sergei Vasilievich Ivanov was able to prove an analogue of Novikov–Adian theorem: for any m > 1 and an even n ≥ 248, n divisible by 29, the group B(m, n) is infinite. Both Novikov–Adian and Ivanov 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 groups, and there exist non-cyclic finite subgroups. Moreover, the word
and conjugacy
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 Burnside's problem is formed by finitely generated non-cyclic infinite groups in which every nontrivial proper subgroup is a finite cyclic group
, 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
p (one can take p > 1075) of a finitely generated infinite group in which every nontrivial proper subgroup is a cyclic group
of order p. In a paper published in 1996, Ivanov and Ol'shanskii solved an analogue of Burnside's problem in an arbitrary hyperbolic group
for sufficiently large exponents.
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
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
of B(m, n) (otherwise, there exists a subgroup g -1Mg with finite index containing elements not in M). One can therefore define a group B0(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 B0(m,n).
The restricted Burnside problem then asks whether B0(m,n) is a finite group.
In the case of the prime exponent p, this problem was extensively studied by A. I. Kostrikin during the 1950s, prior to the negative solution of the general Burnside problem. His solution, establishing the finiteness of B0(m,p), used a relation with deep questions about identities in Lie algebra
s in finite characteristic. The case of arbitrary exponent has been completely settled in the affirmative by Efim Zelmanov
, who was awarded the Fields Medal
in 1994 for his work.
William Burnside
William Burnside was an English mathematician. He is known mostly as an early contributor to the theory of finite groups....
in 1902 and one of the oldest and most influential questions in group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, asks whether a finitely generated
Generating set of a group
In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...
group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
in which every element has finite order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
must necessarily be a finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
. In plain language, if by looking at individual elements of a group we suspect that the whole group is finite, must it indeed be true? The problem has many variants (see bounded and restricted below) that differ in the additional conditions imposed on the orders of the group elements.
Brief history
Initial work pointed towards the affirmative answer. For example, if a group G is generated by m elements and the order of each element of G is a divisor of 4, then G is finite. Moreover, A. I. Kostrikin (for the case of a prime exponent) and Efim ZelmanovEfim Zelmanov
Efim Isaakovich Zelmanov is a Russian mathematician, known for his work on combinatorial problems in nonassociative algebra and group theory, including his solution of the restricted Burnside problem. He was awarded a Fields Medal at the International Congress of Mathematicians in Zürich in...
(in general) proved that, among the finite groups with given number of generators and exponent, there exists a largest one. Issai Schur
Issai Schur
Issai Schur was a mathematician who worked in Germany for most of his life. He studied at Berlin...
showed that any finitely generated periodic group that was a subgroup of the group of invertible n x n complex matrices was finite; he used this theorem to prove the Jordan–Schur theorem
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...
.
Nevertheless, the general answer to Burnside's 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 and Sergei Adian
Sergei Adian
Sergei Ivanovich Adian, also Adjan is one of the most prominent Soviet and Russian mathematicians. He is a professor at the Moscow State University. He is most famous for his work in group theory, especially on the Burnside problem.-Biography:...
's 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 1010), 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's problem for hyperbolic group
Hyperbolic group
In group theory, a hyperbolic group, also known as a word hyperbolic group, Gromov hyperbolic group, negatively curved group is a finitely generated group equipped with a word metric satisfying certain properties characteristic of hyperbolic geometry. The notion of a hyperbolic group was introduced...
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 periodicPeriodic group
In group theory, a periodic group or a torsion group is a group in which each element has finite order. All finite groups are periodic. The concept of a periodic group should not be confused with that of a cyclic group, although all finite cyclic groups are periodic.The exponent of a periodic group...
if every element has finite order; in other words, for each g in G, there exists some positive integer n such that gn = 1. Clearly, every finite group is periodic. There exist easily defined groups such as the p∞-group
Prüfer group
In mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p∞-group, Z, for a prime number p is the unique p-group in which every element has p pth roots. The group is named after Heinz Prüfer...
which are infinite periodic groups; but the latter group cannot be finitely generated.
The general Burnside problem can be posed as follows:
- If G is a periodic group, and G is finitely generated, then must G necessarily be a finite group?
This question was answered in the negative in 1964 by Evgeny Golod
Evgeny Golod
Evgenii Solomonovich Golod is a Russian mathematician who proved the Golod–Shafarevich theorem on class field towers. As an application, he gave a negative solution to the Kurosh–Levitzky problem on the nilpotency of finitely generated nil algebras, and so to a weak form of Burnside's problem....
and Igor Shafarevich
Igor Shafarevich
Igor Rostislavovich Shafarevich is a Soviet and Russian mathematician, founder of a school of algebraic number theory and algebraic geometry in the USSR, and a political writer. He was also an important dissident figure under the Soviet regime, a public supporter of Andrei Sakharov's Human Rights...
, who gave an example of an infinite p-group
P-group
In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order: each element is of prime power order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power pn is equal to the identity element...
that is finitely generated (see Golod-Shafarevich theorem). However, the orders of the elements of this group are not a priori bounded by a single constant.
Bounded Burnside problem
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. Consider a periodic group G with the additional property that there exists a single integer n such that for all g in G, gn = 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:- 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 x1,…,xm in which the identity xn = 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 g1,…,gm and of exponent n, there is a unique homomorphism from B(m, n) to G that maps the ith generator xi of B(m, n) into the ith generator gi of G. In the language of group presentations
Presentation of a group
In mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of powers of some of these generators, and a set R of relations among those generators...
, free Burnside group B(m, n) has m generators x1,…,xm and the relations xn = 1 for each word x in x1,…,xm, 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 a homomorphic image
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
of B(m, n), where m is the number of generators of G. Burnside's problem can now be restated as follows:
- For which positive integers m, n is the free Burnside group B(m,n) finite?
The full solution to Burnside's problem in this form is not known. Burnside considered some easy cases in his original paper:
- For m = 1 and any positive n, B(1, n) is the cyclic groupCyclic groupIn group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
of order n. - B(m, 2) is the direct product of m copies of the cyclic group of order 2. The key step is to observe that the identities a2 = b2 = (ab)2 = 1 together imply that ab = ba, so that a free Burnside group of exponent two is necessarily abelianAbelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
.
The following additional results are known (Burnside, Sanov, M. Hall
Marshall Hall (mathematician)
Marshall Hall, Jr. was an American mathematician who made contributions to group theory and combinatorics.- Career :...
):
- 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 Burnside's problem was achieved by Pyotr Novikov and Sergei Adian
Sergei Adian
Sergei Ivanovich Adian, also Adjan is one of the most prominent Soviet and Russian mathematicians. He is a professor at the Moscow State University. He is most famous for his work in group theory, especially on the Burnside problem.-Biography:...
in 1968.
Using a complicated combinatorial argument, they demonstrated that for every odd
Even and odd numbers
In mathematics, the parity of an object states whether it is even or odd.This concept begins with integers. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2...
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. The case of even exponent turned out to be considerably more difficult. It was only in 1992 that Sergei Vasilievich Ivanov was able to prove an analogue of Novikov–Adian theorem: for any m > 1 and an even n ≥ 248, n divisible by 29, the group B(m, n) is infinite. Both Novikov–Adian and Ivanov 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 groups, and there exist non-cyclic finite subgroups. Moreover, the word
Word problem for groups
In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two words in the generators represent the same element...
and conjugacy
Conjugacy problem
In abstract algebra, the conjugacy problem for a group G with a given presentation is the decision problem of determining, given two words x and y in G, whether or not they represent conjugate elements of G...
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 Burnside's problem is formed by finitely generated non-cyclic infinite groups in which every nontrivial proper subgroup is a finite cyclic group
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
, the so-called Tarski Monsters
Tarski monster group
In mathematics, a Tarski monster group, named for Alfred Tarski, is an infinite group G, such that every proper subgroup H of G, other than the identity subgroup, is a cyclic group of order a fixed prime number p. A Tarski monster group is necessarily simple. It was shown by A. Yu...
. 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
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
p (one can take p > 1075) of a finitely generated infinite group in which every nontrivial proper subgroup is a cyclic group
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
of order p. In a paper published in 1996, Ivanov and Ol'shanskii solved an analogue of Burnside's problem in an arbitrary hyperbolic group
Hyperbolic group
In group theory, a hyperbolic group, also known as a word hyperbolic group, Gromov hyperbolic group, negatively curved group is a finitely generated group equipped with a word metric satisfying certain properties characteristic of hyperbolic geometry. The notion of a hyperbolic group was introduced...
for sufficiently large exponents.
Restricted Burnside problem
The restricted Burnside problem, formulated in the 1930s, asks another, related, question:- 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 isomorphismGroup isomorphismIn abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic...
?
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 of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
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
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
of B(m, n) (otherwise, there exists a subgroup g -1Mg with finite index containing elements not in M). One can therefore define a group B0(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 B0(m,n).
The restricted Burnside problem then asks whether B0(m,n) is a finite group.
In the case of the prime exponent p, this problem was extensively studied by A. I. Kostrikin during the 1950s, prior to the negative solution of the general Burnside problem. His solution, establishing the finiteness of B0(m,p), used a relation with deep questions about identities in Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
s in finite characteristic. The case of arbitrary exponent has been completely settled in the affirmative by Efim Zelmanov
Efim Zelmanov
Efim Isaakovich Zelmanov is a Russian mathematician, known for his work on combinatorial problems in nonassociative algebra and group theory, including his solution of the restricted Burnside problem. He was awarded a Fields Medal at the International Congress of Mathematicians in Zürich in...
, who was awarded the Fields Medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
in 1994 for his work.
External links
- History of the Burnside Problem at MacTutor History of Mathematics archiveMacTutor History of Mathematics archiveThe 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...