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.

Text sample: Chapter Introduction: […] The importance of Solomons algebra for several combinatorial and algebraical contexts is one cause for Thorsten Bauer to study within his thesis [3] the algebraical structure of Solomons algebra in characteristic zero. Again, one main insight for his study is the description of a second and suitable linear basis. Among other things he describes the stagnation of the ascending and descending Lie-central chains, its derivations and algebra automorphism, the Carter subgroups of its group of units and the Cartan subalgebras of ist associated Lie algebra. The latter ones are studied in a more general context of finite-dimensional associative soluble algebras. Both structures are connected, too: the Carter subgroups are the group of units of the Cartan subalgebras. The research of Thorsten Bauer for Solomons algebra are an additional motivation for me to study the structure of the Solomon-Tits algebra. The study within this work is done based on the following questions: - Is it possible to generalize the results of Thorsten Bauer and to apply them to the Solomon-Tits algebra? - Do more connections between the group of units and the Lie algebra4 of an associative soluble algebra exist? - What are insights about the associative structure, the group-theoretical structure of the group of units and the Lie algebra structure of the associated Lie algebra of Solomon-Tits algebra? - Is it possible to apply the results to Solomons algebra? These questions will not be answered completely within this work but they are the guidelines for our research. Now we present our results and the meaning of self-centralizing radical complements is highlighted as a more general context for studying our guidelines.

• Solomon

• associative algebra

• Lie algebra

• group of units

• radical factor algebra

• Duo algebras

• Tadashi Nakayama

Sven Bodo Wirsing was born in 1975 in Neumünster. After graduating from high school at KKS in Itzehoe (with a focus on mathematics and physics), he studied mathematics with a minor in business administration (especially logistics) at CAU university in Kiel. He did his doctorate in 2005 on group and algebra theory. During his years of study in Kiel he gained experience in the analysis of interdisciplinary processes, which are reflected in different disciplines of algebra, such as group theory, representation theory, theory of Lie and associative algebras. From this experience, he also studied and analyzed the subject matter of the present work. Since the end of his doctorate, Dr. Wirsing has been working as a senior IT consultant for logistics processes at several renowned companies where he is responsible for logistics optimization and maintenance. Since 2012 he has published several books on algebras.