1. A CENTURY OF THE BURNSIDE PROBLEM
J´anos Kurdics kurdics@nyf.hu
Seminar of Institute of Mathematics and Informatics of Ny´ıregyh´aza College
22nd of October, 2015, translated from Hungarian
J´anos Kurdics kurdics@nyf.hu BURNSIDE PROBLEM . . .

2. General Burnside Problem (1902)
Every torsion group is locally finite.
Finite exponent Burnside Problem
Every torsion group of finite exponent is locally finite.
Let F be a free group of rank d, N = Fe. Then B(d, e) = F/N is
the Burnside group of parameter (d, e). The finite exponent
Burnside Problem is equivalent to finiteness of B(d, e)
Restricted Burnside Problem (W. Magnus)
There are finitely many isomorphism classes of finite d-generated
groups of exponent e.
If the restricted Burnside Problem holds, B(d, e) can be infinite.
Finitely many normal divisors of finite index, however, have the
intersection of finite index, hence there is a biggest finite
d-generated group B0(d, e) of exponent e, which has all finite
d-generated groups of exponent e as a homomorphix image.
J´anos Kurdics kurdics@nyf.hu BURNSIDE PROBLEM . . .

3. William Burnside 1852-1927
for full biography visit
http://www-history.mcs.st-and.ac.uk/Biographies/Burnside.html
J´anos Kurdics kurdics@nyf.hu BURNSIDE PROBLEM . . .

4. Parents
Emma Knight and William Burnside Sr
1858
Father died, he attended Christ’s Hospital primary
1871
Grant to Cambridge St John’s College
1873
Pembroke College
1875
Second wrangler to George Chrystal
1875
Smith grant, teaching assitant job in Pembroke, studies
hidrodynamics
J´anos Kurdics kurdics@nyf.hu BURNSIDE PROBLEM . . .

5. 1885
Professorship in Greenwich Royal Navy College
1891-92,94
Studies complex linear fractional function groups, then group
theory
1897
The Theory of Groups of Finite Order
1906-1908
Presides London Mathematical Society
1925
Stroke
1928
The Theory of Probability
J´anos Kurdics kurdics@nyf.hu BURNSIDE PROBLEM . . .

6. Igor Rostislavovich Shafarevich 1923-
for full biography visit
http://www-history.mcs.st-and.ac.uk/Biographies/Shafarevich.html
J´anos Kurdics kurdics@nyf.hu BURNSIDE PROBLEM . . .

7. Parents
Rostislav Stepanovich Shafarevich and Juliya Yakovlevna Vasilyeva
1938
At MGU takes examinations by B.Ny. Delone, A. G. Kurosh and
I.M. Gelfand
1940,42
graduation, candidate of sciences
1944
MGU Mechmat Faculty
1946
Doctor of Sciences, has 423 science descendants
50’s,70’s suspended
J´anos Kurdics kurdics@nyf.hu BURNSIDE PROBLEM . . .

8. Burnside 1902
B(d, 3) is finite, |B(d, 3)| ≤ 32m − 1, B(2, 4) is finite,
|B(2, 4)| ≤ 212
Burnside 1905
Torsion complex matrix group is locally finite.
Sanov 1940
B(d, 4) is finite.
A.I. Kostrikin 1955
B0(2, 5) is finite.
Higman 1956
B0(d, 5) and B0(d, 6) exist.
J´anos Kurdics kurdics@nyf.hu BURNSIDE PROBLEM . . .

9. Marshall Hall 1958
B(d, 6) is finite
Ye.S. Golod and I.R. Shafarevich 1964
There exists a finitely generated infinite dimensional nilalgebra.
There exists a finitely generated infinite p-group, general Burnside
Problem is refuted.
S.I. Adyan ´es P.S. Novikov 1968
B(d, e) is infinite for odd e ≥ 4381 exponent, finite exponent
Burnside Problem is refuted.
Yu. Olshanskiy 1982
For p > 1075 primes there exists infinite groups with all proper
subgroups of order p.
Open problems
A finitely presented torsion group is finite. A finitely presented
algebraic algebra is finite dimensional. B(2, 5) is finite.
J´anos Kurdics kurdics@nyf.hu BURNSIDE PROBLEM . . .

10. Yefim Isakovich Zelmanov 1955-
for full biography visit
http://www-history.mcs.st-and.ac.uk/history/Biographies/Zelmanov.html
J´anos Kurdics kurdics@nyf.hu BURNSIDE PROBLEM . . .

11. 1977,79
Graduation, candidate of sciences at NGU, studies nonassociative
algebras
1980,85
Matinstitute Novosibirsk, doctor of sciences
1990,94
Professor at Wisconsin-Madison and then at Chicago
1994
Fields Medal
1995-2002
Professor at Yale and then at San Diego
J´anos Kurdics kurdics@nyf.hu BURNSIDE PROBLEM . . .

12. Schreier conjecture 1926
Outer automorphism groups of finite simple groups are soluble.
Ph. Hall and G. Higman 1956
Schreier conjecture implies that it suffices to prove the restricted
Burnside Problen for prime power exponents
1984
Classification of finite simple groups settles Schreier conjecture
W. Magnus 1950
For exponent p the restricted Burnside Problem holds provided
ecery p − 1-Engel Lie algebra over Zp is locally nilpotent
J´anos Kurdics kurdics@nyf.hu BURNSIDE PROBLEM . . .

13. 1959 A.P. Kosztrikin
Every p − 1-Engel Lie algebra is locally nilpotent
is
E.I. Zelmanov 1989
For exponents pk reduction to local nilpotency of Engel Lie rings
E.I. Zelmanov 1991
Every Engel Lie ring is locally nilpotent settling the restricted
Burnside Problem
J´anos Kurdics kurdics@nyf.hu BURNSIDE PROBLEM . . .

14. Commutator identities
(x, yz) = (x, z)(x, y)z
(xy, z) = (x, z)y
(y, z) (x, y−1
) = (y, x)y−1
Hall-Witt (x, y−1
, z)y
(y, z−1
, x)z
(z, x−1
, y)x
Our aim is to prove the Burnside problem for exponent 3.
(1) A group is 2-Engel iff every conjugacy class generates an
Abelian normal divisor.
This follows from the identity (x, y, y) = ((y−1)x , y).
(2) A group of exponent 3 is 2-Engel
This follows from (1) and the identity
(xy , x) = ((x−1y)y )3(y−2x)3(yx )3.
J´anos Kurdics kurdics@nyf.hu BURNSIDE PROBLEM . . .

15. (3) In a 2-Engel group (x, y, z) = (y, z, x) = (z, x, y).
We have (x, yz, z) = (x, y, z)y , but by (1) (x, y, z) commutes with
each argument hence (x, yz, z) = (x, y, z), analogously
(x, yz, y) = (x, z, y). In a 2-Engel group (x, y−1) = (y, x). Then
1 = (x, yz, z)(x, yz, y) = (x, y, z)(x, z, y) and (x, y, z) = (z, x, y).
Applying this again to (z, x, y) one gets (z, x, y) = (y, z, x).
(4) In a 2-Engel group, (x, y, z)3 = 1
This follows from Hall-Witt identity and (3).
J´anos Kurdics kurdics@nyf.hu BURNSIDE PROBLEM . . .

16. (5) A 2-Engel group is nilpotent of class at most 3. (In particular,
it is locally finite.)
Applying the identities and (3) we see
(x, y, z, w) = (z, w, (x, y)) = ((w, z), (x, y))−1
= (x, y, w, z)−1
. By similar technique,
(x, y, z, w) = (x, y, w, z)−1
= ((w, x, y), z)−1
= ((w, x), y, z)−1
=
(y, z, (w, x))−1
= ((w, x), (y, z)) = ((y, z), w, x) = (y, z, w, x)
. This yields (y, z, w, x) = (y, z, x, w)−1 = (x, y, z, w)−1 and
(x, y, z, w) = (x, y, z, w)−1 = (x, y, z, w)2 by (4).
J´anos Kurdics kurdics@nyf.hu BURNSIDE PROBLEM . . .

17. References
1 Burnside, W. (1902): On an unsettled question in the theory of
discontinuous groups, Quart. J. Pure and Applied Math. 33 (1902),
230-238.
2 Golod, E.S., Shafarevich, I.R. (1964): On the class field tower, Izv.
Akad. Nauk SSSR Ser. Mat., 28:2, 261–272.
3 Kostrikin, A.I. (1986): Vokrug Bernsayda, Nauka, Moszkva.
4 Kuros, A.G. (1941): Problem¨u tyeorii kol´ec, svyazann¨uye s problem¨u
Bernsayda periodicheskih gruppah, Izv. Akad. Nauk SSSR 5, no. 3,
233–240.
5 O’Connor, J.J., Robertson, F.: MacTutor History of Mathematics,
St. Andrews, http://www-history.mcs.st-and.ac.uk, accessed
12.09.2015
6 Sahoo, B.K., Sury, B. (2005): What is the Burnside problem?
Resonance 10(7), 34-48.
7 Sanov, I.I. (1940): Resenyiye problem¨u Bernsayda dlya pakazatyelya
4, Ucs. Zap. LGU 55, 166-170.
8 Zelmanov, Efim (2007): Some open problems in the theory of
infinite dimensional algebras, J. Korean Math. Soc. 44, No. 5,
1185–1195.
J´anos Kurdics kurdics@nyf.hu BURNSIDE PROBLEM . . .