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The Abstract Group G3,7,16: A Correction
Published online by Cambridge University Press: 20 January 2009
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Dr. A. D. Sands has pointed out that the following sentence, occurring in (2, p. 60, lines 2 to 4), should be deleted: “Hence Sinkov's group of order 1344 … is the holomorph of the Abelian group {pi}, of order 8 (1, pp. 111–117).” In fact, Sinkov's group and the holomorph of C2×C2×C2 are not isomorphic. For, the holomorph is representable on 8 letters by definition (1, p. 87), whereas Sinkov's group is not representable on 8 letters. To see this, we recall that Sinkov's group (3, p. 584) is generated by two elements of periods 2 and 3 (namely, QP3 and QP2) whose commutator is of period 8. If these two generators could be represented as permutations of 8 letters, their commutator would be an even permutation and thus could not be of period 8.
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References
(1) Burnside, W., Theory of Groups of Finite Order (2nd ed., Cambridge University Press, 1911).Google Scholar
(2) Coxeter, H. S. M., The abstract group G3716, these Proceedings, 13 (Series II, 1962), 47–61.Google Scholar
(3) Sinkov, A., On the group-defining relations (2, 3, 7; p). Ann. of Math., (2) 38 (1937), 577–584.CrossRefGoogle Scholar
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