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Schur's exponent conjecture - counterexamples of exponent $5$ and exponent $9$ | ||
| International Journal of Group Theory | ||
| دوره 10، شماره 4، اسفند 2021، صفحه 167-173 اصل مقاله (176.48 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2020.123980.1638 | ||
| نویسنده | ||
| Michael Vaughan-Lee* | ||
| Oxford University Mathematical Institute, United Kingdom | ||
| چکیده | ||
| There is a long-standing conjecture attributed to I. Schur that if $G$ is a finite group with Schur multiplier $M(G)$ then the exponent of $M(G)$ divides the exponent of $G$. In this note I give an example of a four generator group $G$ of order $5^{4122}$ with exponent $5$, where the Schur multiplier $M(G)$ has exponent $25$. | ||
| کلیدواژهها | ||
| Schur multiplier؛ Groups of exponent $5$؛ Schur’s exponent conjecture | ||
| مراجع | ||
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[1] A. J. Bayes, J. Kautsky and J. W. Wamsley, Computation in nilpotent groups (applicaton), Proceedings of the second international conference on the theory of groups (Australian National University, Canberra,1973), Springer, Berlin, 1974 82–89. [2] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput., 24 (1997) 235–265. [3] G. Havas and M. F. Newman, Applications of computers to questions like those of Burnside, Lecture Notes in Mathematics, 806, Berlin, Springer-Verlag, (1980) 211–230. [4] G. Havas, M. F. Newman and M. R. Vaughan-Lee, A nilpotent quotient algorithm for graded Lie rings, J. Symbolic Computation, 9 (1990) 653–664. [5] G. Higman, Some remarks on varieties of groups, Quart. J. Math. Oxford, 10 (1959) 165–178.
[6] V. Thomas, On Schur’s exponent conjecture and its relation to Noether’s rationality problem, arXiv:2007.03476, 2020. | ||
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