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Burnside problem
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What is the Burnside problem in mathematics?
In group theory, the Burnside problem, posed by William Burnside in 1902, questions whether a finitely generated periodic group with elements of finite order is necessarily a finite group.
A group is finitely generated if a finite number of its elements can produce every element in the group through combinations. An element has finite order if its product with itself eventually yields the group's identity element. A periodic group is one in which every element has finite order. The answer to Burnside's question is no, as demonstrated in 1964 by Yevgeny Solomonovich Golod, who constructed an infinite period group using a finite number of generators with finite order.
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Who introduced the Burnside problem and when?
The Burnside problem, formulated in 1902 by English mathematician William Burnside, questions whether a finitely generated periodic group, where each element has a finite order, must be a finite group.
Burnside couldn't solve his original problem, leading him to pose a related question: Are all finitely generated groups of bounded exponent finite? This is known as the bounded Burnside problem. Russian mathematicians Sergei Adian and Petr Novikov resolved the bounded Burnside problem in 1968, demonstrating that the answer was no for all odd (n \geq 4,381). Another variant, the restricted Burnside problem, asks: For fixed positive integers m and n, are there only finitely many groups generated by m elements of bounded exponent n? Efim Isaakovich Zelmanov, a Russian mathematician, affirmatively answered the restricted Burnside problem, earning a Fields Medal in 1994.
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What is a 'group' in mathematics, as related to the Burnside problem?
In mathematics, specifically in the context of the Burnside problem, a group is a set of elements together with a binary operation that satisfies specific axioms. These axioms are:
- Closure The combination of any two elements results in another element within the group.
- Associativity For elements (a), (b), and (c), the associative law holds: ((a * b) * c = a * (b * c)).
- Identity element There exists an identity element (e) such that for every element (a) in the group, (a * e = a = e * a).
- Inverse element For every element (a), there exists an inverse element (a^{-1}) such that (a * a^{-1} = e = a^{-1} * a).
A group is called commutative or Abelian if it also satisfies the commutative law, meaning that for all elements (a) and (b), (a * b = b * a).
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What question does the Burnside problem ask about groups with certain properties?
The Burnside problem, formulated in 1902 by English mathematician William Burnside, questions whether a finitely generated periodic group, where each element has a finite order, must be a finite group.
Burnside's initial problem was unanswered, leading to a related question known as the bounded Burnside problem: Are all finitely generated groups of bounded exponent finite? This considers the order, or exponent, for each element.
Another variant Burnside pondered was the restricted Burnside problem: For fixed positive integers m and n, are there only finitely many groups generated by m elements of bounded exponent n?
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How did mathematicians attempt to answer the Burnside problem?
The Burnside problem, posed in 1902 by English mathematician William Burnside, questions whether a finitely generated periodic group is necessarily a finite group. A group is finitely generated if a finite number of its elements can produce every element in the group. Each element in a periodic group has a finite order.
Yevgeny Solomonovich Golod, a Russian mathematician, disproved Burnside's original problem in 1964 by constructing an infinite period group with a finite number of generators with finite order.
Burnside then posed a related question: Are all finitely generated groups of bounded exponent finite? Mathematicians Sergei Adian and Petr Novikov resolved this problem in 1968 by proving the answer was no for all odd (n \geq 4,381). In 1994, Efim Isaakovich Zelmanov won a Fields Medal for his affirmative answer to the restricted Burnside problem.
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What are some important discoveries or solutions related to the Burnside problem?
The Burnside problem, posed in 1902 by English mathematician William Burnside, questions whether a finitely generated periodic group is necessarily finite.
Russian mathematician Yevgeny Solomonovich Golod disproved Burnside's original problem in 1964 by constructing an infinite period group with a finite number of generators with finite order.
Burnside then posed a related question: Are all finitely generated groups of bounded exponent finite? In 1968, Russian mathematicians Sergei Adian and Petr Novikov resolved this bounded Burnside problem, showing the answer was no for odd n ≥ 4,381. The lower bound has since decreased to all odd n ≥ 665 and then for all n ≥ 8,000.
For fixed positive integers m and n, the restricted Burnside problem asks: Are there only finitely many groups generated by m elements of bounded exponent n? Russian mathematician Efim Isaakovich Zelmanov won a Fields Medal in 1994 for his affirmative answer to the restricted Burnside problem.
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Burnside problem, in group theory (a branch of modern algebra), problem of determining if a finitely generated periodic group with each element of finite order must necessarily be a finite group. The problem was formulated by the English mathematician William Burnside in 1902.
A finitely generated group is one in which a finite number of elements within the group suffice to produce through their combinations every element in the group. For example, all the positive integers (1, 2, 3…) can be generated using the first element, 1, by repeatedly adding it to itself. An element has finite order if its product with itself eventually produces the identity element for the group. An example is the distinct rotations and “flip overs” of a square that leave it oriented the same way in the plane (i.e., not tilted or twisted). The group then consists of eight distinct elements, all of which can be generated by various combinations of just two operations: a 90° rotation and a flip. The dihedral group, as it is called, therefore needs just two generators, and each generator has finite order; four 90° rotations or two flips return the square to its the original orientation. A periodic group is one in which each element has finite order. It was clear to Burnside that an infinite group (such as the positive integers) may have a finite number of generators and a finite group must have finite generators, but he wondered if every finitely generated periodic group must necessarily be finite. The answer turned out to be no, as shown in 1964 by the Russian mathematician Yevgeny Solomonovich Golod, who was able to construct an infinite period group using only a finite number of generators with finite order.
Burnside was unable to answer his original problem, so he asked a related question: Are all finitely generated groups of bounded exponent finite? Known as the bounded Burnside problem, the distinction has to do with the order, or exponent, for each element. For example, Golod’s group did not have a bounded exponent; that is, it did not have a single number n such that, for any element in the group, g ∊G, gn = 1 (where 1 indicates the identity element rather than necessarily the number 1). Russian mathematicians Sergei Adian and Petr Novikov in 1968 resolved the bounded Burnside problem by showing that the answer was no, for all odd n ≥ 4,381. Through the decades since Burnside pondered the problem, the lower bound has decreased, first by Adian in 1975 to all odd n ≥ 665 and finally in 1996 by the Russian mathematician I.G. Lysenok for all n ≥ 8,000.
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Meanwhile, Burnside had pondered yet another variant, known as the restricted Burnside problem: For fixed positive integers m and n, are there are only finitely many groups generated by m elements of bounded exponent n? The Russian mathematician Efim Isaakovich Zelmanov was awarded a Fields Medal in 1994 for his affirmative answer to the restricted Burnside problem. Various other conditions considered by Burnside are still areas of active mathematical research.