On the structure of the Solomon-Tits algebra of the symmetric group. An analysis of associative, group theoretic and Lie theoretical phenomenons

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Within this work we study the structure of the Solomon-Tits algebra of the symmetric group motivated by results of research done by Manfred Schocker about the module structure of this algebra. We investigate three structures: the associative, the associated Lie algebra and the group of units. All three structures are related and can be studied in the more general context of associative soluble splittable algebras possessing a self-centralizing radical complement. Our results are related to dimension formulas, Duo algebras, self-centralization of the radical complements, Cartan subalgebras, Sylow subgroups, Hall subgroups, Carter subgroups, stagnation of central chains, classes of nilpotency and solvability, exponents alongside central chains, nilradical and Fitting subgroup, semisimple and simple substructures, anti-automoprhism and irreducible character values.