# Paper by I. N. Sanov, Solution of the Burnside problem for exponent 4

I have searched extensively online and for copies of printed journals containing the paper which details Sanov's solution to the Burnside Problem for exponent 4, which is widely cited in many papers and texts on the Burnside problem.

The reference for the paper is I. N. Sanov, "Solution of the Burnside problem for exponent 4", Uchen. Zap. Leningrad State Univ. Ser. Mat. 10 (1940), 166-170 (i.e. Leningrad State University Annals).

• I haven't seen Sanov's paper, but it appears that Marshall Hall's book "The Theory of Groups" (AMS Chelsea Publishing 1959) contains an account of the proof in Section 18.3. I can show you if interested. Commented Dec 5, 2014 at 18:00
• You may find the proof of the local finiteness of groups of exponent four in Derek J. S. Robinson's group theory book published by Springer. Commented Dec 6, 2014 at 18:33
• Both are excellent suggestions, thanks. Commented Dec 8, 2014 at 14:53

As it was pointed out by Alex Dugas, one can find Sanov theorem in Hall's book here, see Theorem 18.3.1. According to the author, the proof does not determine precisely the order of $B(n,4)$. However, it is quite easy to show that $B(2,4)$ is finite: $|B(2,4)|=4096$.

Let me add that the following GAP code verifies that $|B(2,4)|\leq4096$. (Then it is easy to conclude that indeed one has $|B(2,4)|=4096$.) The idea is to generate a random set $w_1,w_2,\dots,w_N$ for some big $N$ and checks whether the group $F_2/\langle w_1^4,...,w_N^4\rangle$ is finite:

gap> f := FreeGroup(2);;
gap> a := f.1;;
gap> b := f.2;;
gap> rels := Set(List([1..10000],x->Random(f)^4));;
gap> gr := f/rels;;
gap> Order(gr);
4096

• There exist very short presentations for $B(2,4)$, like the one with 9 relators, see my answer below: mathoverflow.net/a/445712/2164 Commented Apr 28, 2023 at 15:33

Burnside proved in [1] that the order of $$B(2,4)$$ is $$\le4096$$. Tobin proved in [4] that this number is the correct order for this group. A short presentation for $$B(2,4)$$ was given by Leech in [3], which is reproduced in Coxeter-Moser's book [2]: $$B(2,4)=\langle a,b \mid a^4, b^4, (ab)^4, (a^{-1}b)^4, (ab^2)^4, (a^2b)^4, (a^{-1}b^{-1}ab)^4, (a^2b^2)^4, (a^{-1}bab)^4, (ab^{-1}ab)^4\rangle$$ Any one of the last four relators can be omitted.

UPDATE 5/23/23: Interestingly, the first 6 relators are of the form $$w^4$$, where $$w$$ is some primitive element of the free group on $$a,b$$ (which means that $$w$$ is part of some basis of the free group $$F(a,b)$$). Whereas the other 4 relators are of the form $$u^4$$ where $$u$$ is not primitive.

References:

[1] Burnside, W. On an unsettled question in the theory of discontinuous groups. Quart. J. 33, 230-238 (1902).

[2] Coxeter, H. S. M.; Moser, W. O. J. Generators and relations for discrete groups. Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 14. Springer-Verlag, Berlin-New York, 1980.

[3] Leech, John. Coset enumeration on digital computers. Proc. Cambridge Philos. Soc. 59 (1963), 257–267.

[4] Tobin, John Joseph. On Groups with Exponent Four. Thesis (Ph.D.)–The University of Manchester (United Kingdom). 1954. 107 pp.

• “Any of the last four relators can be omitted.” Do you mean any one of the last four? Or do you mean that all can be omitted? (I guess the former based on your comment above, but it is ambiguous) Commented Apr 28, 2023 at 15:37
• @Carl-FredrikNybergBrodda thank you for this notice, I meant "any one". Corrected now. Commented Apr 28, 2023 at 15:42