A new approach to character-free proof for Frobenius theorem
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-si...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Amirkabir University of Technology
2023-02-01
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| Series: | AUT Journal of Mathematics and Computing |
| Subjects: | |
| Online Access: | https://ajmc.aut.ac.ir/article_4863_36ef014a4f85c1bbee9f6f358b602a34.pdf |
| Summary: | 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. |
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| ISSN: | 2783-2449 2783-2287 |