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For basic definitions and results of group theory please see a
standard text such as [2] or [8].
In a finite group *G* every element *g* satisfies *g*^{n} = *e* for some least
positive integer *n* called the order of *g* and denoted by `o`(*g*).
This leads us to,

**Problem 1** (General Burnside Problem)
Let *G* be a finitely generated group such that for every element *g* of
*G* there is a positive integer *N*_{g} so that *g*^{Ng} = *e*. Then is *G*
finite?

When *G* arises as a group of *n*×*n* matrices (or more formally
when *G* is a linear group) it was shown by Burnside that the answer is
yes (a simple proof is outlined in Appendix A). However, in 1964
Golod and Shafarevich [3]
showed that this is not true for all groups.
Thereafter, Alyoshin [1], Suschansky [12],
Grigorchuk [4] and Gupta-Sidki [5]
gave various counter-examples.
We can tighten the above conjecture since we know that, `o`(*g*) divides
`o`(*G*) the number of elements of the set of elements of *G*. Thus we can
formulate,

**Problem 2** (Ordinary Burnside Problem)
Let *G* be a finitely generated group for which there is a positive integer
*N* such that for every element *g* we have *g*^{N} = *e*. Then is *G* finite?

(We call the smallest such integer *N* the *exponent* of *G*.)
In 1968 Novikov and Adian [10] gave counter-examples
for groups of odd exponents for the Ordinary Burnside Problem.
If we are primarily interested only in *finite* groups and their
classification then we can again restrict the problem further. Thus
Magnus [9] formulated the following problem.

**Problem 3** (Restricted Burnside Problem)
Is there a finite number *A*(*k*, *N*) of *finite* groups which are
generated by *k* elements and have exponent *N*?

Alternatively one can ask if the order of all such groups is
uniformly bounded. Hall and Higman [6]
proved that the Restricted
Burnside problem for a number *N* which can be factorised as
*p*_{1}^{n1 ... }*p*_{r}^{nr} follows from the following three hypothesis:
- The Restricted Burnside Problem is true for
*p*_{i}^{ni}.
- There are at most finitely many finite simple group
quotients which are
*k*-generated and have exponent *N*.
- For each finite simple group quotient
*G* as above the
group of outer automorphisms of *G* is a solvable group.

Thus modulo the latter two problems which have to do with the
Classification of Finite Simple groups, we reduced to a study of the
Restricted Burnside Problem for *p*-groups. We note that a key step in
the Classification of Finite Simple groups was the celebrated theorem
of Feit and Thompson which won a Fields Medal in 1970. According
to this theorem if *N* is odd then there are no finite simple groups
in items (2) and (3) above.

** Next:** 3 p-Groups
** Up:** The Work of Efim
** Previous:** 1 Introduction
Kapil Hari Paranjape
2002-11-22