Editing Burnside problem (section)

== Restricted Burnside problem ==
Formulated in the 1930s, it asks another, related, question:

<blockquote>'''Restricted Burnside problem.''' If it is known that a group ''G'' with ''m'' generators and exponent ''n'' is finite, can one conclude that the order of ''G'' is bounded by some constant depending only on ''m'' and ''n''? Equivalently, are there only finitely many ''finite'' groups with ''m'' generators of exponent ''n'', up to [[group isomorphism|isomorphism]]?</blockquote>

This variant of the Burnside problem can also be stated in terms of category theory: an affirmative answer for all ''m'' is equivalent to saying that the category of finite groups of exponent ''n'' has all finite limits and colimits.<ref name="Nahlus-Yang">{{cite arXiv |eprint=2107.09900 |page=19 |last1=Nahlus |first1=Nazih |last2=Yang |first2=Yilong |title=Projective Limits and Ultraproducts of Nonabelian Finite Groups |date=2021 |class=math.GR }} Corollary 3.2</ref> 
It can also be stated more explicitly in terms of certain universal groups with ''m'' generators and exponent ''n''. By basic results of group theory, the intersection of two [[normal subgroups]] of finite [[Index of a subgroup|index]] in any group is itself a normal subgroup of finite index. Thus, the intersection ''M'' of all the normal subgroups of the free Burnside group B(''m'', ''n'') which have finite index is a normal subgroup of B(''m'', ''n''). One can therefore define the free restricted Burnside group B<sub>0</sub>(''m'', ''n'') to be the [[quotient group]] B(''m'', ''n'')/''M''. Every finite group of exponent ''n'' with ''m'' generators is isomorphic to B(''m'',''n'')/''N'' where ''N'' is a normal subgroup of B(''m'',''n'') with finite index. Therefore, by the [[isomorphism theorem | Third Isomorphism Theorem]], every finite group of exponent ''n'' with ''m'' generators is isomorphic to B<sub>0</sub>(''m'',''n'')/(''N''/''M'') — in other words, it is a homomorphic image of B<sub>0</sub>(''m'', ''n'').
The restricted Burnside problem then asks whether B<sub>0</sub>(''m'', ''n'') is a finite group. 
In terms of category theory, B<sub>0</sub>(''m'', ''n'')  is the coproduct of ''n'' cyclic groups of order ''m'' in the category of finite groups of exponent ''n''. 

In the case of the prime exponent ''p'', this problem was extensively studied by [[A. I. Kostrikin]] during the 1950s, prior to the negative solution of the general Burnside problem. His solution, establishing the finiteness of B<sub>0</sub>(''m'', ''p''), used a relation with deep questions about identities in [[Lie algebra]]s in finite characteristic. The case of arbitrary exponent has been completely settled in the affirmative by [[Efim Zelmanov]], who was awarded the [[Fields Medal]] in 1994 for his work.