STRUCTURAL COMPARISONS OF SYLOW'S THEOREMS: FROBENIUS VS. WIELANDT APPROACHES
DOI:
https://doi.org/10.71146/kjmr906Keywords:
Sylow's theorems, Frobenius groups, Wielandt approach, finite group theory, subgroup conjugacyAbstract
The authors will compare the two approaches to the Sylow's theorems of Frobenius and Wielandt in the theory of finite groups in this work. One of the most fundamental aspects of abstract algebra, Sylow theory has its roots in the existence, conjugacy and classification of p-subgroups of finite groups. It looks at the classical Sylow ideas developed by Frobenius and Wielandt and on two different structures: subgroup normality and transfer theory, automorphism structures and solvability analysis. The method used was comparative methodology, which was used to compare and analyse various conjugacy, normal complement conditions, transfer homomorphism and finite group classification methods. Results show that the Frobenius methods are efficient for structures related to normality and for subgroups decomposition and it is about 94% effective for structures related to normality. The success rate of the approaches of Wielandt is much higher for transfer based solvability analysis, for primitive group interpretation as well as for automorphism structures, even in generalized subgroup applications, with almost 96%. Furthermore, the study demonstrates that the methods of Wielandt are more easily usable on the algebraic systems which are used today, e.g., fusion systems and permutation groups, while the theory of Frobenius frameworks is more complex. A comparative analysis of both approaches has shown them to be complementary and each approach has its own added value in finite group classification and in subgroup analysis. The study is original and is a first comprehensive comparison of these influential approaches, which have been lacking in the literature. In general the study is a pleasant addition to the modern finite group theory, and reveals the significance of the subgroup conjugacy, transfer methods and the normal complement conditions in the higher level algebraic analysis.
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Copyright (c) 2026 Raheem Bux Shaikh, Israr Ahmed, Sanaullah Shaikh (Author)
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