Solution of the Burnside Problem for Exponent 6

Hall, Marshall

doi:10.1073/pnas.43.8.751

Cited by 26 publications

(30 citation statements)

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“…Burnside proved that B(m, 3) is finite for all m, and also proved that 5(2,4) is finite. In 1940 Sanov [37] proved that B(m, 4) is finite for all m, and in 1958 Hall [14] proved that B(m, 6) is finite for all m. To date, no other Burnside groups (apart from the cyclic Burnside groups) are known to be finite, although a great deal of work has been done in an attempt to determine whether or not B (2,5) is finite. In the other direction, Novikov and Adjan [33][34][35] proved that B(m, n) is infinite if m > 1 and n is odd and n > 4381.…”

Section: A Still Undecided Point In the Theory Of Discontinuous Groupmentioning

confidence: 99%

Bounds in the restricted Burnside problem

Vaughan-Lee¹,

Zelmanov²

1999

J Aust Math Soc A

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Section: A Still Undecided Point In the Theory Of Discontinuous Groupmentioning

confidence: 99%

Bounds in the restricted Burnside problem

Vaughan-Lee¹,

Zelmanov²

1999

J Aust Math Soc A

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“…The group B(r, n) is finite when r = 1, or r is an arbitrary positive integer and n = 2, 3, 4, 6 (see [4,19,29,9]). It was proved by Novikov and Adjan ([21,22,23]) that B(r, n) is infinite when r > 1, n is odd, and n ≥ 4381.…”

Section: Definition 37mentioning

confidence: 99%

Turing degrees of nonabelian groups

Dabkowska

Da̧bkowski

Harizanov

et al. 2007

Proc. Amer. Math. Soc.

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Abstract. For a countable structure A, the (Turing) degree spectrum of A is the set of all Turing degrees of its isomorphic copies. If the degree spectrum of A has the least degree d, then we say that d is the (Turing) degree of the isomorphism type of A. So far, degrees of the isomorphism types have been studied for abelian and metabelian groups. Here, we focus on highly nonabelian groups. We show that there are various centerless groups whose isomorphism types have arbitrary Turing degrees. We also show that there are various centerless groups whose isomorphism types do not have Turing degrees.

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“…But a group of exponent 6 is soluble: its finitely generated subgroups are finite and consequently soluble of bounded derived length by the results of [3]. Since G is simple it contains elements of order 5 which do not commute and so (viii) G contains a subgroup H isomorphic to PSL (2,5).…”

Section: Any Two Non-commuting Elements Of G Of Order 5 Generate a Sumentioning

confidence: 99%

Two-variable laws forPSL(2, 5)

Bryant

Powell

1969

J. Aust. Math. Soc.

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In [2], John Cossey and Sheila Oates Macdonald give a basis for the set of laws of PSL(2, 5) — the simple group of order 60 — and with one extreme exception the laws of their basis involve at most two variables. They raise the problem of finding a basis in which all of the laws involve only a small number of variables, and remark that they have shown that five variables will suffice. Here we give a basis consisting entirely of two variable laws.

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Solution of the Burnside Problem for Exponent 6

Cited by 26 publications

References 0 publications

Bounds in the restricted Burnside problem

Bounds in the restricted Burnside problem

Turing degrees of nonabelian groups

Two-variable laws forPSL(2, 5)

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