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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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4
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OFFSET
| 0,2
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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.
B(1,r), B(2,r), B(3,r), B(4,r) and B(6,r) are all finite: |B(1,r)| = r, |B(2,r)| = 2^r, |B(3,r)| = A051576, |B(4,r)| = A079682, |B(6,r)| = A079683.
B(e,r) is infinite for e > 2 and n >= 13 (Ivanov).
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REFERENCES
| M. Hall, Jr., The Theory of Groups, Macmillan, 1959, Chap. 18.
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.
W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Wiley, 1966, see p. 380.
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LINKS
| J. J. O'Connor and E. F. Robertson, History of the Burnside Problem
D. Rusin, Burnside Problem
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FORMULA
| 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.
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CROSSREFS
| Sequence in context: A072240 A182795 A182796 * A115528 A106226 A005070
Adjacent sequences: A079680 A079681 A079682 * A079684 A079685 A079686
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KEYWORD
| nonn,bref
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jan 31 2003
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EXTENSIONS
| The next term is too large to include.
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