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For our purposes a *p*-group is a finite group such that it has *p*^{a}
elements for some non-negative integer *a*. The Restricted Burnside
problem for such groups can be stated as follows.

**Problem 4** (Restricted Burnside Problem for *p*-groups)
Let *G* be a finite group with exponent *p*^{n} which is generated by *k*
elements. Then *G* has *p*^{a} elements for some integer *a*. Is there a
uniform bound *a*(*k*, *n*) for *a*?

To study *p*-groups we first note that these are nilpotent.
We define the Central series for *G*
*G*_{1} = *G* and by induction on *i*, *G*_{i} = [*G*, *G*_{i - 1}]

Recall that *G* is *nilpotent* if *G*_{i} is the trivial group of
order 1 for some *i*. Now if *G* is a *p*-group then the abelian groups
*G*_{i}/*G*_{i + 1} have order a power of *p*. Thus we can construct a finer
series called the *p*-Central series for *G* a *p*-group
*G*_{1} = *G* and*G*_{i + 1} is the subgroup generated by[*G*, *G*_{i}] and the set*G*_{i}^{p}

By the above discussion it follows that *G*_{i} becomes trivial for
large enough *i*; in addition, each
*G*_{i}/*G*_{i + 1} is a vector space
over
*Z*/*p**Z* for all smaller *i*. The
*Z*/*p**Z*-vector space
*L*(*G*) is defined as
*L*(

*G*) =

*G*_{i}/

*G*_{i + 1}
The non-commutative structure of *G* can be caught by a Lie algebra
structure on *L*(*G*). We recall the definition of a Lie algebra.

**Definition 1**
A vector space

*L* over a field

*k* is said to be a

*lie algebra*
if there is a pairing
[,] :

*L*×

*L**L* with the following
properties

[*x*, *y*] = - [*y*, *x*] and [*x*,[*y*, *z*]] + [*z*,[*x*, *y*]] + [*y*,[*z*, *x*]] = 0

The Lie algebra structure on *L*(*G*) is given by
*G*_{i}/

*G*_{i + 1}×

*G*_{j}/

*G*_{j + 1}*G*_{i + j}/

*G*_{i + j + 1}
where the map is
(,) (Check
that this is well-defined!).
The above lie algebra has some additional structure. First is an
identity proved by Higman [7]. If *G* has exponent *p*^{n} then

ad(

*a*_{(1)})

`o`ad(

*a*_{(2)})

`o`^{ ... }`o`ad(

*a*_{(pn - 1)}) = 0

as a map
*L*(*G*)*L*(*G*); here
ad(*a*) : *L**L* for any element *a* in a
Lie algebra is the map
*b* [*a*, *b*].
The second identity is proved by Sanov [11]. Let *x*_{i} be the
elements of
*G*_{0}/*G*_{1} *L*(*G*) corresponding to the finitely many
generators *g*_{i} of *G*. Then for any a commutator on the *x*_{i}
we have
ad()^{pn} = 0.

The main result of Zelmanov can formulated as follows.

**Theorem 1** (Zelmanov)
Let

*L* be

*any* Lie algebra over

*Z*/

*p**Z* which is generated as
a Lie algebra by

*k* elements

*x*_{i} such that we have the Higman and
Sanov identities.Then

*L* is nilpotent as a Lie algebra.

The interested reader can find this proof outlined in [13].

** Next:** 4 The Proof of
** Up:** The Work of Efim
** Previous:** 2 Groups
Kapil Hari Paranjape
2002-11-22