Burnside group

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 1902, 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. also Burnside problem).

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 1933, F. Levi and B.L. van der Waerden (see [a4]) proved that the Burnside group has the class of nilpotency equal to , when , and the order equals , where are binomial coefficients. In 1940, I.N. Sanov [a18] proved that the free Burnside groups of exponent are also finite. In 1954, S.J. Tobin proved that (see [a4]). By making use of computers, A.J. Bayes, J. Kautsky, and J.W. Wamsley showed in 1974 that and W.A. Alford, G. Havas and M.F. Newman established in 1975 that (see [a4]). It is also known (see [a4]) that the class of nilpotency of equals when . On the other hand, in 1978, Yu.P. Razmyslov constructed an example of a non-solvable countable group of exponent (see [a4]). In 1958, 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 1950 and asks whether there exists a number so that the order of any finite -generator group of exponent is less than . The existence of such a bound was proven for prime by A.I. Kostrikin [a11] in 1959 (see also [a12]) and for with a prime number by E.I. Zel'manov [a19], [a20] in 1991–1992. 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 1968, P.S. Novikov and S.I. 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. also Identity problem; Conjugate elements).

A much simpler construction of free Burnside groups for and odd was given by A.Yu. Ol'shanskii [a16] in 1982 (see also [a17]). In 1994, further developing Ol'shanskii's geometric method, S.V. 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.

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.G. Lysenok [a13] to in 1996). 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 . It is also shown in [a9] that the word and conjugacy problems for the constructed presentation of are solvable. In fact, these decision problems are effectively reduced to the word problem for groups and it is shown that each satisfies a linear isoperimetric inequality and hence is a Gromov hyperbolic group [a5] (cf. Hyperbolic group).

It should be noted that the structure of finite subgroups of the groups , is very complex when the exponent is even and, in fact, finite subgroups of , play a key role in proofs in [a9] (which, like [a15], also contains a large number of lemmas, proved by simultaneous induction). The central result related to finite subgroups of the groups , is the following: Let , where is the maximal odd divisor of . Then any finite subgroup of , is isomorphic to a subgroup of the direct product for some , where denotes a dihedral group of order . The principal difference between odd and even exponents in the Burnside problem can be illustrated by pointing out that, on the one hand, for every odd there are infinite -generator groups of exponent all of whose proper subgroups are cyclic (as was proved in [a2], see also [a17]) and, on the other hand, any -group the orders of whose Abelian (or finite) subgroups are bounded is itself finite (see [a7]).

In 1997, Ivanov and Ol'shanskii [a10] showed that the above description of finite subgroups in is complete (that is, every subgroup of can actually be found in ) and obtained the following result: Let be a finite -subgroup of . Then the centralizer of in contains a subgroup isomorphic to a free Burnside group of infinite countable rank such that , whence . (Since obviously contains subgroups isomorphic to both and , an embedding of in becomes trivial.) Among other results on subgroups of proven in [a10] are the following: The centralizer of a subgroup is infinite if and only if is a locally finite -group. Any infinite locally finite subgroup is contained in a unique maximal locally finite subgroup while any finite -subgroup is contained in continuously many pairwise non-isomorphic maximal locally finite subgroups. A complete description of infinite (maximal) locally finite subgroups of has also been obtained, in [a10].

References

[a1]	S.I. Adian, "The Burnside problems and identities in groups" , Springer (1979) (In Russian)
[a2]	V.S. Atabekian, S.V. Ivanov, "Two remarks on groups of bounded exponent" , 2243-B87 , VINITI, Moscow (1987) ((This is kept in the Depot of VINITI, Moscow, and is available upon request))
[a3]	W. Burnside, "An unsettled question in the theory of discontinuous groups" Quart. J. Pure Appl. Math. , 33 (1902) pp. 230–238
[a4]	N. Gupta, "On groups in which every element has finite order" Amer. Math. Monthly , 96 (1989) pp. 297–308
[a5]	M. Gromov, "Hyperbolic groups" S.M. Gersten (ed.) , Essays in Group Theory , Springer (1987) pp. 75–263
[a6]	Ph. Hall, G. Higman, "On the -length of -soluble groups and reduction theorems for Burnside's problem" Proc. London Math. Soc. , 6 (1956) pp. 1–42
[a7]	D. Held, "On abelian subgroups of an infinite -group" Acta Sci. Math. (Szeged) , 27 (1966) pp. 97–98
[a8]	M. Hall Jr., "Solution of the Burnside problem for exponent " Proc. Nat. Acad. Sci. USA , 43 (1957) pp. 751–753
[a9]	S.V. Ivanov, "The free Burnside groups of sufficiently large exponents" Internat. J. Algebra Comput. , 4 (1994) pp. 1–308
[a10]	S.V. Ivanov, A.Yu. Ol'shanskii, "On finite and locally finite subgroups of free Burnside groups of large even exponents" J. Algebra , 195 (1997) pp. 241–284
[a11]	A.I. Kostrikin, "On the Burnside problem" Math. USSR Izv. , 23 (1959) pp. 3–34 (In Russian)
[a12]	A.I. Kostrikin, "Around Burnside" , Nauka (1986) (In Russian)
[a13]	I.G. Lysenok, "Infinite Burnside groups of even period" Math. Ross. Izv. , 60 (1996) pp. 3–224
[a14]	W. Magnus, "A connection between the Baker–Hausdorff formula and a problem of Burnside" Ann. Math. , 52 (1950) pp. 11–26 (Also: 57 (1953), 606)
[a15]	P.S. Novikov, S.I. Adian, "On infinite periodic groups I–III" Math. USSR Izv. , 32 (1968) pp. 212–244; 251–524; 709–731
[a16]	A.Yu. Ol'shanskii, "On the Novikov–Adian theorem" Math. USSR Sb. , 118 (1982) pp. 203–235 (In Russian)
[a17]	A.Yu. Ol'shanskii, "Geometry of defining relations in groups" , Kluwer Acad. Publ. (1991) (In Russian)
[a18]	I.N. Sanov, "Solution of the Burnside problem for exponent " Uch. Zapiski Leningrad State Univ. Ser. Mat. , 10 (1940) pp. 166–170
[a19]	E.I. Zel'manov, "Solution of the restricted Burnside problem for groups of odd exponent" Math. USSR Izv. , 36 (1991) pp. 41–60 (In Russian)
[a20]	E.I. Zel'manov, "A solution of the restricted Burnside problem for -groups" Math. USSR Sb. , 72 (1992) pp. 543–565 (In Russian)