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Search: id:A051576
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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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+0
4
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| 1, 3, 27, 2187, 4782969, 847288609443, 36472996377170786403, 1144561273430837494885949696427, 78551672112789411833022577315290546060373041, 35370553733215749514562618584237555997034634776827523327290883
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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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.
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
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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
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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
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a(n) = 3^{n*(n^2+5)/6} for n >= 0.
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CROSSREFS
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Equals 3^A004006(n).
Sequence in context: A009066 A012012 A102580 this_sequence A158113 A078233 A009039
Adjacent sequences: A051573 A051574 A051575 this_sequence A051577 A051578 A051579
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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