OFFSET
0,2
COMMENTS
The Burnside group of exponent e and rank r is B(e,r) := F_r / N where F_r is the free group generated by x_1, ..., x_r and N is the normal subgroup generated by all z^e with z in F_r. The Burnside problem is to determine when B(e,r) is finite. [Warning: Some authors interchange the order of e and r. But the symbol is not symmetric. B(i,j) != B(j,i). - N. J. A. Sloane, Jan 12 2016]
B(1,r), B(2,r), B(3,r), B(4,r) and B(6,r) are all finite: |B(1,r)| = 1, |B(2,r)| = 2^r, |B(3,r)| = A051576, |B(4,r)| = A079682, |B(6,r)| = A079683. |B(5,2)| = 5^34.
Many cases are known where B(e,r) is infinite (see references). Ivanov showed in 1994 that B(e,r) is infinite if r>1, e >= 2^48 and 2^9 divides e if e is even.
It is not known whether B(5,2) is finite or infinite.
The next term, a(3), is 2^4375*3^833. - N. J. A. Sloane, Jan 12 2016
See A051576 for additional references.
REFERENCES
M. Hall, Jr., The Theory of Groups, Macmillan, 1959, Chap. 18.
W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Wiley, 1966, see p. 380.
LINKS
S. V. Ivanov, On the Burnside problem for groups of even exponent, Proc. Internat. Congress of Mathematicians, Vol. II (Berlin, 1998). Doc. Math. 1998, Extra Vol. II, 67-75.
J. J. O'Connor and E. F. Robertson, History of the Burnside Problem
FORMULA
The formula for a(n) was found by Marshall Hall, Jr.: a(n) = 2^i 3^(j + (j choose 2) + (j choose 3)) where i = 1 + (n-1)3^(n + (n choose 2) + (n choose 3)) and j = 1 + (n-1)2^n. (See also the Maple code.)
MAPLE
B6n:=proc(n) local a, b, c;
b:=1+(n-1)*2^n;
c:=n+binomial(n, 2)+binomial(n, 3);
a:=1+(n-1)*3^c;
2^a*3^(b+binomial(b, 2)+binomial(b, 3));
end; # N. J. A. Sloane, Jan 12 2016
CROSSREFS
KEYWORD
nonn,bref
AUTHOR
N. J. A. Sloane, Jan 31 2003
EXTENSIONS
Entry revised by N. J. A. Sloane, Jan 12 2016 and Jan 15 2016
STATUS
approved