Dipl. Math. Felix F. Flemisch Libri

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Part 3 of the second Trilogy „The Strong Sylow Theorem for the Prime p in the Locally Finite Classical Groups“ & „The Strong Sylow Theorem for the Prime p in Locally Finite and p-Soluble Groups“ & "Augustin-Louis Cauchy's and Évariste Galois' Contributions to Sylow Theory in Finite Groups" proves for a subgroup G of the finite group H Lagrange's theorem and three group theorems by Cauchy, where the second and the third were concealed, by a unified method of proof consisting in smart arranging the elements of H resp. the cosets of G in H in a rectangle/tableau. Cauchy's third theorem requires the existence of a Sylow p-subgroup of H. These classical proofs are supplemented by modern proofs based on cosets resp. double cosets which take only a few lines. We then analyse first his well-known published group theorem of 1845/1846, for which he constructs a Sylow p-subgroup of Sn, thereby correcting a misunderstanding in the literature and introducing wreath products, and second his published group theorem of 1812/1815, which is related to theorems of Lagrange, Vandermonde and Ruffini. Subsequently we present what Galois knew about Cauchy's group theorems and about Sylow's theorems by referring to his published papers and as well to his posthumously published papers and to his manuscripts. We close with a detailed narrative of early group theory and early Sylow theory in finite groups.

In Part 3 of the Trilogy „Characterising Locally Finite Groups Satisfying the Strong Sylow Theorem for the Prime p“ & „About the Strong Sylow Theorem for the Prime p in Simple Locally Finite Groups“ & „The Strong Sylow Theorem for the Prime p in Projective Special Linear Locally Finite Groups“ we continue the program begun in [10] to optimise along the way 1) its beautiful Theorem about the first type „An“ of infinite families of finite simple groups step-by-step to further types by proving it for the second type „A = PSL n“. We start with proving the Conjecture 2 of [10] about the General Linear Groups over (commutative) locally finite fields, stating that their rank is bounded in terms of their p-uniqueness, and then break down this insight to the Special Linear Groups and the Projective Special Linear (PSL) Groups over locally finite fields. We close with suggestions for future research regarding the remaining five rank-unbounded types (the „Classical Groups“) and the way 2), regarding the (locally) finite and p-soluble groups, and regarding Augustin-Louis Cauchy's and Évariste Galois' contributions to Sylow theory in finite groups.