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A051576
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Order of Burnside group B(3,n) of exponent 3 and rank n.
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4
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1, 3, 27, 2187, 4782969, 847288609443, 36472996377170786403, 1144561273430837494885949696427, 78551672112789411833022577315290546060373041, 35370553733215749514562618584237555997034634776827523327290883
(list; graph; refs; listen; history; internal format)
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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(5,2)| = 5^34.
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
C. C. Sims, Concerning B(5,2)
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FORMULA
| a(n) = 3^{n*(n^2+5)/6} for n >= 0.
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MATHEMATICA
| 3^Table[n*(n^2 + 5)/6, {n, 0, 10}] (* From Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)
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CROSSREFS
| Equals 3^A004006(n).
Sequence in context: A192341 A191511 A102580 * A184278 A158113 A078233
Adjacent sequences: A051573 A051574 A051575 * A051577 A051578 A051579
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KEYWORD
| nonn,easy,nice,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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