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A079683
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Order of Burnside group B(6,n) of exponent 6 and rank n.
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3
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OFFSET
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0,2
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COMMENTS
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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.
See A051576 for additional references.
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REFERENCES
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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.
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LINKS
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FORMULA
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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.)
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MAPLE
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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));
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CROSSREFS
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KEYWORD
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nonn,bref
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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