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A new approach to character-free proof for Frobenius theorem | ||
| AUT Journal of Mathematics and Computing | ||
| مقاله 12، دوره 4، شماره 1، 2023، صفحه 99-103 اصل مقاله (330.43 K) | ||
| نوع مقاله: Original Article | ||
| شناسه دیجیتال (DOI): 10.22060/ajmc.2022.21305.1085 | ||
| نویسندگان | ||
| Seyedeh Fatemeh Arfaeezarandi* 1؛ Vahid Shahverdi2 | ||
| 1Department of Mathematics, Stony Brook University, Stony Brook, New York, USA | ||
| 2Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden | ||
| چکیده | ||
| Let G be a Frobenius group. Using character theory, it is proved that the Frobenius kernel of G is a normal subgroup of G, which is well-known as a Frobenius theorem. There is no known character-free proof for Frobenius theorem. In this note, we prove it, by assuming that Frobenius groups are non-simple. Also, we prove that whether K is a subgroup of G or not, Sylow 2-subgroups of G are either cyclic or generalized quaternion group. Also by assuming some additional arithmetical hypothesis on G we prove Frobenius theorem. We should mention that our proof is character-free. | ||
| کلیدواژه‌ها | ||
| Finite group؛ Frobenius group؛ Frobenius theorem | ||
| مراجع | ||
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[9] K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math. Phys., 3 (1892), pp. 265–284. | ||
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