论文信息 - Pseudo Sylow Numbers

Pseudo Sylow Numbers

Abstract One part of Sylow’s famous theorem in group theory states that the number of Sylow p-subgroups of a finite group is always congruent to 1 modulo p. Conversely, Marshall Hall has shown that not every positive integer occurs as the number of Sylow p-subgroups of some finite group. While Hall’s proof relies on deep knowledge of modular representation theory, we show by elementary means that no finite group has exactly 35 Sylow 17-subgroups.

Benjamin Sambale

[1] Benjamin Sambale. Pseudo Frobenius numbers , 2018, Expositiones Mathematicae.

[2] J. LaFountain. Inc. , 2013, American Art.

[3] The early proofs of Sylow's theorem , 1980 .

[4] M. L. Sylow. Théorèmes sur les groupes de substitutions , 1872 .

[5] Thomas Wieting Reed,et al. THE THEOREM OF FROBENIUS , 2001 .

[6] I. Isaacs. Algebra, a graduate course , 1994 .

[7] M. Hall. On the number of Sylow subgroups in a finite group , 1967 .

[8] Michael Aschbacher,et al. The Status of the Classification of the Finite Simple Groups , 2004 .

[9] N. Sloane. The on-line encyclopedia of integer sequences , 2018, Notices of the American Mathematical Society.

[10] P. Hall. A Note on Soluble Groups , 1928 .

[11] J. Brodkey. A note on finite groups with an abelian Sylow group , 1963 .

[12] W. Feit,et al. SOLVABILITY OF GROUPS OF ODD ORDER , 2012 .

[13] R. Tennant. Algebra , 1941, Nature.

[14] Geoffrey R. Robinson. Cauchy's Theorem and Sylow's Theorem from the Cyclic Case , 2011, Am. Math. Mon..