Burnside problem

The Burnside problem posed by William Burnside in 1902 is one of the oldest and most influential problems in group theory and asks whether every finitely generated group in which every element has finite order must necessarily be finite . Yevgeny Golod and Igor Schafarewitsch found a counterexample in 1964. The problem has several variants, such as the restricted Burnside problem or the restricted Burnside problem , which differ in that the element orders are subject to further conditions.

Historical remarks

At first everything pointed to a positive answer. For example, if the group is finitely generated and every element order is a factor of 4, then is already finite. In 1958 AI Kostrikin was able to show that among the finite groups with a given number of producers and with a given prime number as the group exponent, there is one largest. This gives a solution to the constrained Burnside problem in the case of a prime exponent. Later, in 1989, Efim Zelmanov was able to solve the restricted Burnside problem for any exponent. In 1911 Issai Schur showed that every finitely generated torsion group contained in the complex, invertible n  × n matrices must be finite. He used this to prove the Jordan-Schur theorem . ${\displaystyle G}$ ${\displaystyle G}$

That said, the Burnside problem turned out to have a negative answer. In 1964, Golod and Schafarewitsch found an infinite group of the Burnside type (i.e. finitely generated and all elements have a finite order) without the element orders being equally restricted. In 1968 Pyotr Novikow and Sergei Adjan provided a negative solution to the problem with bounded exponents for all odd exponents greater than 4381. In 1982 Alexander Olschanski found some startling counterexamples for sufficiently large odd exponents (greater than ) and, based on geometrical ideas, provided one considerable simpler proof. ${\displaystyle 10^{10}}$

The case of even exponents has been found to be much more difficult. In 1992 SW Ivanov announced a negative solution for sufficiently large straight exponents that are divisible by a large power of two . The full evidence was published in 1994 and was a good 300 pages. Later joint work with Olschanski led to a negative solution of the analogue of the Burnside problem for hyperbolic groups , again for sufficiently large exponents. In contrast, for small exponents other than 2, 3, 4 or 6, almost nothing is known.

The general Burnside problem

A group is called a torsion group if every element has a finite order, that is, if there is a positive integer for every element such that . Obviously every finite group is a torsion group. There are infinite torsion groups that are easy to define, such as the examiner groups , but the latter are not finite. ${\displaystyle G}$ ${\displaystyle g\in G}$ ${\displaystyle n}$ ${\displaystyle g^{n}=1}$

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

The answer to this question was negative, in 1964 Yevgeny Golod and Igor Schafarewitsch found an example of an infinite, finitely generated p group . The element orders of this group were not restricted by a common constant.

The limited Burnside problem

The Cayley graph of the 27-element free Burnside group with 2 generators and an exponent 3

Part of the difficulties with the general Burnside problem arises from the fact that the requirements of the finite generation and the torsion group allow only little information about the possible group structure. Therefore we need further demands . We therefore consider torsion groups with the additional property that there is a smallest number such that for all . Such groups are called torsion groups with limited exponent or simply groups with exponent . The limited Burnside problem poses the question ${\displaystyle G}$ ${\displaystyle n>0}$ ${\displaystyle g^{n}=1}$ ${\displaystyle g\in G}$ ${\displaystyle n}$ ${\displaystyle n}$

Bounded Burnside Problem: Is a finitely generated torsion group with an exponent necessarily finite? ${\displaystyle n}$

The problem can be reformulated as a question about the finiteness of a particular family of groups. The free Burnside group of rank and exponents , with is called, is a group of producers , in which all elements of the relationship meet, and in some ways most is group with these characteristics. More precisely, it is characterized by the fact that there is exactly one group homomorphism for every other group with producers and with exponent , which maps the -th producer from to the -th producer . In the language of the presentation of a group , the group with generators and relations for each word is in , each additional group with generators and exponents is obtained by requiring additional relations. The existence of the free Burnside groups and their uniqueness result from standard techniques of group theory. So if a group generated by elements is exponent , then there is a surjective group homomorphism . So the constrained Burnside problem can be rephrased as follows: ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle B(m,n)}$ ${\displaystyle m}$ ${\displaystyle x_{1},\ldots ,x_{m}}$ ${\displaystyle x}$ ${\displaystyle x^{n}=1}$ ${\displaystyle B(m,n)}$ ${\displaystyle G}$ ${\displaystyle g_{1},\ldots ,g_{m}}$ ${\displaystyle n}$ ${\displaystyle B(m,n)\rightarrow G}$ ${\displaystyle i}$ ${\displaystyle x_{i}}$ ${\displaystyle B(m,n)}$ ${\displaystyle i}$ ${\displaystyle g_{i}\in G}$ ${\displaystyle B(m,n)}$ ${\displaystyle m}$ ${\displaystyle x_{1},\ldots ,x_{m}}$ ${\displaystyle x^{n}=1}$ ${\displaystyle x}$ ${\displaystyle x_{1},\ldots ,x_{m}}$ ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle G}$ ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle B(m,n)\rightarrow G}$

Restricted Burnside Problem II: For which positive numbers and is the free Burnside group finite? ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle B(m,n)}$

The complete solution to the Burnside problem in this form is unknown. Burnside had considered a few simple cases in his original paper:

• ${\displaystyle B(1,n)}$is the cyclic group of order . ${\displaystyle n}$
• ${\displaystyle B(m,2)}$is the direct product of copies of order 2. (The essential point is the observation that the relations imply the equation , so that free Burnside groups with exponent 2 are Abelian .) ${\displaystyle m}$ ${\displaystyle a^{2}=b^{2}=(ab)^{2}=1}$ ${\displaystyle ab=ba}$

The following results are still known (Burnside, IN Sanow, M. Hall ):

• ${\displaystyle B(m,3)}$, and are finally for everyone . ${\displaystyle B(m,4)}$ ${\displaystyle B(m,6)}$ ${\displaystyle m}$

The special case is open as of 2005, it is not known whether this group is finite. In 1968 Pyotr Novikov and Sergei Adjan achieved a breakthrough in solving the Burnside problem. Using a complicated combinatorial argument, they showed that for every odd number there are infinite, finitely generated groups with exponents . Adjan later improved this to odd exponents greater than 665. (In 1973, John Britton proposed an alternative proof, about 300 pages long, in which Adjan found an error.) The case of even exponents turned out to be considerably more difficult. It was only in 1994 that SW Ivanov was able to prove an analogue to the Novikow-Adjan theorem: For and and divisible by is infinite. Together with the Novikow-Adjan theorem, one can conclude that there are infinite Burnside groups for everyone and . This was improved in 1996 by Igor Gerontjewitsch Lyssjonok (Lysënok) . Novikow-Adjan, Ivanov and Lysënok achieved considerably more precise results on the structure of free Burnside groups. In the case of odd exponents, all finite subsets of the free Burnside group are cyclic. For even exponents, every finite subgroup is contained in a product of two dihedral groups and there are non-cyclic finite subgroups. Moreover, the word and conjugation problems for both even and odd exponents are effectively solvable. ${\displaystyle B(2,5)}$ ${\displaystyle n>4381}$ ${\displaystyle n}$ ${\displaystyle m>1}$ ${\displaystyle n\geq 2^{48}}$ ${\displaystyle n}$ ${\displaystyle 2^{9}}$ ${\displaystyle B(m,n)}$ ${\displaystyle m>1}$ ${\displaystyle n\geq 2^{48}}$ ${\displaystyle n\geq 8000}$ ${\displaystyle B(m,n)}$

A famous class of counterexamples to the Burnside problem are finitely generated, non-cyclic, infinite groups, in which every real subgroup is a finite cyclic group, these are the so-called Tarski groups . The first examples of such groups were constructed in 1979 by AJ Olschanski using geometric methods. He was able to tighten this further in 1982 and show the existence of finitely generated, infinite groups for every sufficiently large prime number in which every non-trivial subgroup is cyclically of order . In a paper published in 1996, Ivanov and Olschanski were able to solve the analogue of the Burnside problem for any hyperbolic group with sufficiently large exponents. ${\displaystyle p>10^{75}}$ ${\displaystyle p}$

The limited Burnside problem

The question from the 1930s is:

Restricted Burnside problem: If a group is known to be generated by elements, has the exponent , and is finite, can one estimate the group order against a constant that is only dependent on and ? That is equivalent to the question: Are there only finitely many finite groups with generators and exponents apart from isomorphism ? ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle m}$ ${\displaystyle n}$

This variant of the Burnside problem can be reformulated into a statement about certain universal groups with generators and exponents . It is a fundamental statement of group theory that the intersection of two subgroups with a finite index again has a finite index. Let be the intersection of all subgroups of with finite index. Then there is a normal divisor (because all conjugates of subgroups with finite index are of this form again) and one can define it as the quotient group. Every finite group with producers and exponent is a homomorphic picture of . The constrained Burnside problem asks if is finite. ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle M}$ ${\displaystyle B(m,n)}$ ${\displaystyle M}$ ${\displaystyle B_{0}(m,n)}$ ${\displaystyle B(m,n)/M}$ ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle B_{0}(m,n)}$ ${\displaystyle B_{0}(m,n)}$

Even before the negative answer to the general Burnside problem, the case of a prime number exponent was studied extensively by AI Kostrikin in the 1950s . Its solution, that is, the finiteness of , used relationships to deep-seated questions about identities in Lie algebras of finite characteristics . The case of general exponents was finally positively solved by Efim Zelmanov , for which he was awarded the Fields Medal in 1994. ${\displaystyle p}$ ${\displaystyle B_{0}(m,p)}$

Individual evidence

1. ^ AI Kostrikin: Around Burnside . Translated from Russian with a foreword by James Wiegold Results of mathematics and its border areas. 3rd episode, volume 20, Springer-Verlag 1990, ISBN 3-540-50602-0 .
2. ^ A History of the Burnside problem in the MacTutor History of Mathematics archive .
3. ^ DJS Robinson : A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , Chapter 14.2: Torsion Groups and the Burnside Problems.
4. ↑ Charles W. Curtis , Irving Reiner : Representation Theory of Finite Groups and Associative Algebras , AMS Chelsea Publishing (1962), ISBN 0-8218-4066-5 , Theorem (36,14).
5. ^ SV Ivanov: The free Burnside groups of sufficiently large exponents , Internat. J. Algebra Comput. (1994), Volume 4, pages 1-308.
6. ↑ ^{a b} S. V. Ivanov, A. Yu. Ol'shanskii: Hyperbolic groups and their quotients of bounded exponents , Trans. Amer. Math. Soc. (1996), Volume 348, pp. 2091-2138.
7. ^ DJS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , sentences 14.2.3, 14.2.4, 14.2.6.
8. ^ SV Ivanov: On the Burnside Problem on Periodic Groups , Bulletin (New Series) of the Amer. Math. Soc. (1992), Volume 27.2, pp. 257-260 (review).
9. ↑ IG Lysënok: Infinite Burnside groups with straight exponent. Izv. Horse. Akad. Nauk Ser. Mat. (1996) Volume 60.3, pages 3-224. English translation IG Lysënok: Infinite Burnside groups of even exponent , Izv. Math. 60.3, pages 453-654.
10. ^ DJS Robinson : A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , sentence 1.3.12.