### Word Problems: Decision Problems and the Burnside Problem in Group Theory

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Let be a free group of rank. The free -generator Burnside group of exponent is defined to be the quotient group of by the subgroup of generated by all th powers of elements of. Clearly, is the "largest" -generator group of exponent that is, a group whose elements satisfy the identity in the sense that if is an -generator group of exponent then there exists an epimorphism. In , W. Burnside [a3] posed a problem which later became known as the Burnside problem for periodic groups that asks whether every finitely-generated group of exponent is finite, or, equivalently, whether the free Burnside groups are finite cf.

## Word problems. Decision problems and the Burnside problem in group theory

It is easy to show that the free -generator Burnside group of exponent is an elementary Abelian 2-group and the order of is. Burnside showed that the groups are finite for all. In , F. Levi and B. In , I. Sanov [a18] proved that the free Burnside groups of exponent are also finite. In , S.

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Tobin proved that see [a4]. By making use of computers, A. Bayes, J. Kautsky, and J. Wamsley showed in that and W. Alford, G. Havas and M.

## Word problems: Decision problems and the Burnside problem in group theory

Newman established in that see [a4]. It is also known see [a4] that the class of nilpotency of equals when. On the other hand, in , Yu. Razmyslov constructed an example of a non-solvable countable group of exponent see [a4]. In , M. Hall [a8] proved that the Burnside groups of exponent are finite and have the order given by the formula , where and. The attempts to approach the Burnside problem via finite groups gave rise to a restricted version of the Burnside problem called the restricted Burnside problem which was stated by W.

Magnus [a14] in and asks whether there exists a number so that the order of any finite -generator group of exponent is less than.

## Book Word Problems Decision Problems And The Burnside Problem In Group Theory

The existence of such a bound was proven for prime by A. Kostrikin [a11] in see also [a12] and for with a prime number by E. Zel'manov [a19] , [a20] in — It then follows from the Hall—Higman reduction results [a6] and the classification of finite simple groups that a bound does exist for all and. In , P.

Novikov and S. Adyan [a15] gave a negative solution to the Burnside problem for sufficiently large odd exponents by an explicit construction of infinite free Burnside groups , where and is odd, , by means of generators and defining relators. See [a15] for a powerful calculus of periodic words and a large number of lemmas, proved by simultaneous induction.

Later, Adyan [a1] improved on the estimate for the exponent and brought it down to odd.

Using their machinery, Novikov and Adyan obtained other results on the free Burnside groups. In particular, the word and conjugacy problems were proved to be solvable for the presentations of constructed in [a15] , any Abelian or finite subgroup of was shown to be cyclic for these and other results, see [a1] ; cf.

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A much simpler construction of free Burnside groups for and odd was given by A. Ol'shanskii [a16] in see also [a17]. In , further developing Ol'shanskii's geometric method, S. Ivanov [a9] constructed infinite free Burnside groups , where , and is divisible by if is even, thus providing a negative solution to the Burnside problem for almost all exponents.

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The construction of free Burnside groups given in [a16] , [a9] is based on the following inductive definitions. Let be a free group over an alphabet , , let and let be divisible by from now on these restrictions on and are assumed, unless otherwise stated; note that this estimate was improved on by I. Lysenok [a13] to in By induction on , let and, assuming that the group with is already constructed as a quotient group of , define to be a shortest element of if any the order of whose image under the natural epimorphism is infinite. Then is constructed as a quotient group of by the normal closure of.

Clearly, has a presentation of the form , where are the defining relators of. It is proven in [a9] and in [a16] for odd that for every the word does exist. Furthermore, it is shown in [a9] and in [a16] for odd that the direct limit of the groups as obtained by imposing on of relators for all is exactly the free -generator Burnside group of exponent. The infiniteness of the group already follows from the existence of the word for every , since, otherwise, could be given by finitely many relators and so would fail to exist for sufficiently large. Citations Publications citing this paper.

www.cantinesanpancrazio.it/components/kikonum/3-cosa-controllare-prima.php Computable groups which do not embed into groups with decidable conjugacy problem Arman Darbinyan. Asymptotic invariants, complexity of groups and related problems Mark V. Algorithmic and asymptotic properties of groups Mark V. References Publications referenced by this paper. Conjugacy problem and Higman embeddings. Olshanskii , M. Combinatorial Group Theory R. Lyndon , Paul E. A course paper. Ilya Belyaev.

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