Burnside group

Let $F _ { m }$ be a free group of rank $m$. The free $m$-generator Burnside group $B ( m , n )$ of exponent $n$ is defined to be the quotient group of $F _ { m }$ by the subgroup $F _ { m } ^ { n }$ of $F _ { m }$ generated by all $n$th powers of elements of $F _ { m }$. Clearly, $B ( m , n )$ is the "largest" $m$-generator group of exponent $n$ (that is, a group whose elements satisfy the identity $x ^ { n } \equiv 1$) in the sense that if $G$ is an $m$-generator group of exponent $n$ then there exists an epimorphism $\phi : B ( m , n ) \rightarrow G$. 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 $n$ is finite, or, equivalently, whether the free Burnside groups $B ( m , n )$ are finite (cf. also Burnside problem).

It is easy to show that the free $m$-generator Burnside group $B ( m , 2 )$ of exponent $2$ is an elementary Abelian 2-group and the order $| B ( m , 2 ) |$ of $B ( m , 2 )$ is $2 ^ { m }$. Burnside showed that the groups $B ( m , 3 )$ are finite for all $m$. In 1933, F. Levi and B.L. van der Waerden (see [a4]) proved that the Burnside group $B ( m , 3 )$ has the class of nilpotency equal to $3$, when $m > 3$, and the order $| B ( m , 3 ) |$ equals $3 ^ { C _ { m} ^ { 1 } + C _ { m } ^ { 2 } + C _ { m } ^ { 3 } }$, where $C _ { m } ^ { 1 } , \ldots$ are binomial coefficients. In 1940, I.N. Sanov [a18] proved that the free Burnside groups $B ( m , 4 )$ of exponent $4$ are also finite. In 1954, S.J. Tobin proved that $| B ( 2,4 ) | = 2 ^ { 12 }$ (see [a4]). By making use of computers, A.J. Bayes, J. Kautsky, and J.W. Wamsley showed in 1974 that $| B ( 3,4 ) | = 2 ^ { 69 }$ and W.A. Alford, G. Havas and M.F. Newman established in 1975 that $| B ( 4,4 ) | = 2 ^ { 422 }$ (see [a4]). It is also known (see [a4]) that the class of nilpotency of $B ( m , 4 )$ equals $3 m - 2$ when $m > 3$. On the other hand, in 1978, Yu.P. Razmyslov constructed an example of a non-solvable countable group of exponent $4$ (see [a4]). In 1958, M. Hall [a8] proved that the Burnside groups $B ( m , 6 )$ of exponent $6$ are finite and have the order given by the formula $| B ( m , 6 ) | = 2 ^ { \alpha } 3 ^ { C _ { \beta } ^ { 1 } + C _ { \beta } ^ { 2 } + C _ { \beta } ^ { 3 } }$, where $\alpha = 1 + ( m - 1 ) 3 ^ { C _ { m } ^ { 1 } + C _ { m } ^ { 2 } + C _ { m } ^ { 3 } }$ and $\beta = 1 + ( m - 1 ) 2 ^ { m }$.

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 $f ( m , n )$ so that the order of any finite $m$-generator group of exponent $n$ is less than $f ( m , n )$. The existence of such a bound $f ( m , n )$ was proven for prime $n$ by A.I. Kostrikin [a11] in 1959 (see also [a12]) and for $n = p ^ { \ell }$ with a prime number $p$ 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 $f ( m , n )$ does exist for all $m$ and $n$.

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 $B ( m , n )$, where $m \geq 2$ and $n$ is odd, $n\geq 4381$, 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 $n$ and brought it down to odd $n\geq 665$. Using their machinery, Novikov and Adyan obtained other results on the free Burnside groups $B ( m , n )$. In particular, the word and conjugacy problems were proved to be solvable for the presentations of $B ( m , n )$ constructed in [a15], any Abelian or finite subgroup of $B ( m , n )$ 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 $B ( m , n )$ for $m > 1$ and odd $n > 10 ^ { 10 }$ 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 $B ( m , n )$, where $m > 1$, $n \geq 2 ^ { 48 }$ and $n$ is divisible by $2 ^ { 9 }$ if $n$ is even, thus providing a negative solution to the Burnside problem for almost all exponents.

The construction of free Burnside groups $B ( m , n )$ given in [a16], [a9] is based on the following inductive definitions. Let $F _ { m }$ be a free group over an alphabet $\mathcal{A} = \{ a _ { 1 } ^ { \pm 1 } , \ldots , a _ { m } ^ { \pm 1 } \}$, $m > 1$, let $n \geq 2 ^ { 48 }$ and let $n$ be divisible by $2 ^ { 9 }$ (from now on these restrictions on $m$ and $n$ are assumed, unless otherwise stated; note that this estimate $n \geq 2 ^ { 48 }$ was improved on by I.G. Lysenok [a13] to $n \geq 2 ^ { 13 }$ in 1996). By induction on $i$, let $B ( m , n , 0 ) = F _ { m }$ and, assuming that the group $B ( m , n , i - 1 )$ with $i \geq 1$ is already constructed as a quotient group of $F _ { m }$, define $A_i$ to be a shortest element of $F _ { m }$ (if any) the order of whose image (under the natural epimorphism $\psi _ { i - 1 } : F _ { m } \rightarrow B ( m , n , i - 1 )$) is infinite. Then $B ( m , n , i )$ is constructed as a quotient group of $B ( m , n , i - 1 )$ by the normal closure of $\psi _ { i - 1 } ( A _ { i } ^ { n } )$. Clearly, $B ( m , n , i )$ has a presentation of the form $B ( m , n , i ) = \langle a _ { 1 } , \dots , a _ { m } | A _ { 1 } ^ { n } , \dots , A _ { i } ^ { n } \rangle$, where $A _ { 1 } ^ { n } , \dots , A _ { i } ^ { n }$ are the defining relators of $B ( m , n , i )$. It is proven in [a9] (and in [a16] for odd $n > 10 ^ { 10 }$) that for every $i$ the word $A_i$ does exist. Furthermore, it is shown in [a9] (and in [a16] for odd $n > 10 ^ { 10 }$) that the direct limit $B ( m , n , \infty )$ of the groups $B ( m , n , i )$ as $i \rightarrow \infty$ (obtained by imposing on $F _ { m }$ of relators $A_{i}^{n}$ for all $i = 1,2 , \dots$) is exactly the free $m$-generator Burnside group $B ( m , n )$ of exponent $n$. The infiniteness of the group $B ( m , n )$ already follows from the existence of the word $A_i$ for every $i \geq 1$, since, otherwise, $B ( m , n )$ could be given by finitely many relators and so $A_i$ would fail to exist for sufficiently large $i$. It is also shown in [a9] that the word and conjugacy problems for the constructed presentation of $B ( m , n )$ are solvable. In fact, these decision problems are effectively reduced to the word problem for groups $B ( m , n , i )$ and it is shown that each $B ( m , n , i )$ satisfies a linear isoperimetric inequality and hence $B ( m , n , i )$ is a Gromov hyperbolic group [a5] (cf. Hyperbolic group).

It should be noted that the structure of finite subgroups of the groups $B ( m , n , i )$, $B ( m , n )$ is very complex when the exponent $n$ is even and, in fact, finite subgroups of $B ( m , n , i )$, $B ( m , n )$ 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 $B ( m , n , i )$, $B ( m , n )$ is the following: Let $n = n _ { 1 } n _ { 2 }$, where $n_ 1$ is the maximal odd divisor of $n$. Then any finite subgroup $G$ of $B ( m , n , i )$, $B ( m , n )$ is isomorphic to a subgroup of the direct product $D ( 2 n_1 ) \times D ( 2 n_2 ) ^ { \ell }$ for some $\ell$, where $D ( 2 k )$ denotes a dihedral group of order $2 k$. 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 $n \gg 1$ there are infinite $2$-generator groups of exponent $n$ all of whose proper subgroups are cyclic (as was proved in [a2], see also [a17]) and, on the other hand, any $2$-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 $B ( m , n )$ is complete (that is, every subgroup of $D ( 2 n_1 ) \times D ( 2 n_2 ) ^ { \ell }$ can actually be found in $B ( m , n )$) and obtained the following result: Let $G$ be a finite $2$-subgroup of $B ( m , n )$. Then the centralizer $C_{B ( m , n )} ( G )$ of $G$ in $B ( m , n )$ contains a subgroup $B$ isomorphic to a free Burnside group $B ( \infty , n )$ of infinite countable rank such that $G \cap B = \{ 1 \}$, whence $\langle G , B \rangle = G \times B$. (Since $B ( \infty , n )$ obviously contains subgroups isomorphic to both $D ( 2 n _ { 1 } )$ and $D ( 2 n _ { 2 } )$, an embedding of $D ( 2 n_1 ) \times D ( 2 n_2 ) ^ { \ell }$ in $B ( m , n )$ becomes trivial.) Among other results on subgroups of $B ( m , n )$ proven in [a10] are the following: The centralizer $C _ { B ( m , n ) } ( S )$ of a subgroup $S$ is infinite if and only if $S$ is a locally finite $2$-group. Any infinite locally finite subgroup $L$ is contained in a unique maximal locally finite subgroup while any finite $2$-subgroup is contained in continuously many pairwise non-isomorphic maximal locally finite subgroups. A complete description of infinite (maximal) locally finite subgroups of $B ( m , n )$ 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 $p$-length of $p$-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 $2$-group" Acta Sci. Math. (Szeged) , 27 (1966) pp. 97–98
[a8]	M. Hall Jr., "Solution of the Burnside problem for exponent $6$" 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 $4$" 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 $2$-groups" Math. USSR Sb. , 72 (1992) pp. 543–565 (In Russian)