A completely decomposable group is a direct sum of subgroups of the rationals. An almost completely decomposable group is a torsion free abelian group that contains a completely decomposable group as subgroup of finite index. Tight subgroups are maximal subgroups (with respect to set inclusion) among the completely decomposable subgroups of an almost completely decomposable group. In this dissertation we show an extended version of the theorem of Bezout, give a new criterion for the tightness of a completely decomposable subgroup, derive some conditions under which a tight subgroup is regulating and generalize a theorem of Campagna. We give an example of an almost completely decomposable group, all of whose regulating subgroups do not have a quotient with minimal exponent. We show that among the types of elements of a coset modulo a completely decomposable group there exists a unique maximal type and define this type to be -the- coset type. We give criteria for tightness and regulating in term of coset types as well as a representation of the type subgroups using coset types. We introduce the notion of reducible cosets and show their key role for transitions from one completely decomposable subgroup up to another one containing the first one as a proper subgroup. We give an example of a tight, but not regulating subgroup which contains the regulator. We develop the notion of a fully single covered subset of a lattice, show that V-free implies fully single covered, but not necessarily vice versa, and we define an equivalence relation on the set of all finite subsets of a given lattice. We develop some extension of ordinary Hasse diagrams, and apply the lattice theoretic results on the lattice of types and almost completely decomposable groups. / Eine vollständig zerlegbare Gruppe ist eine direkte Summe von Untergruppen der rationalen Zahlen. Eine fast vollständig zerlegbare Gruppe ist eine torsionsfreie abelsche Gruppe, die eine vollständig zerlegbare Gruppe als Untergruppe von endlichem Index enthält. Tight Untergruppen sind bezüglich Mengeninklusion maximale Elemente der Menge der vollständig zerlegbaren Untergruppen einer fast vollständig zerlegbaren Gruppe. In dieser Dissertation zeigen wir eine erweiterte Version des Satzes von Bezout, geben ein neues Kriterium an, mit dem festgestellt werden kann, ob eine Untergruppe tight ist, leiten daraus einige Bedingungen ab, unter denen eine tight Untergruppe regulierend ist, und verallgemeinern einen Satz von Campagna. Wir geben ein Beispiel einer fast vollständig zerlegbaren Gruppe an, deren sämtliche regulierende Untergruppen nicht minimalen Exponenten des Quotienten haben. Wir zeigen, daß unter den Typen der Elemente einer Nebenklasse modulo einer vollständig zerlegbaren Gruppe ein eindeutig definierter maximaler Typ existiert und nennen diesen Typen -den- Nebenklassentypen. Wir geben Kriterien für tight und regulierend mit Hilfe von Nebenklassentypen, sowie eine Darstellung der Typenuntergruppen. Wir führen den Begriff der reduziblen Nebenklassen ein und zeigen die Schlüsselrolle, die diese beim Übergang von einer vollständig zerlegbaren Untergruppe zu einer anderen, die die erste enthält, haben. Wir geben ein Beispiel einer tight Untergruppe an, die nicht regulierend ist, aber den Regulator enthält. Wir führen den Begriff einer "fully single covered" Untermenge eines Verbandes ein, zeigen daß V-frei "fully single covered" impliziert, aber nicht umgekehrt, und definieren eine Äquivalenzrelation auf der Menge aller endlichen Untermengen eines Verbandes. Wir entwickeln eine Erweiterung der üblichen Hasse Diagramme und wenden die verbandstheoretischen Ergebnisse auf die Typenmenge fast vollständig zerlegbarer Gruppen an.

Techn. Hochsch., Diss., 2003--Aachen.

Background Molecular data of histologically classified oligodendrogliomas are available offering the possibility to stratify these human brain tumors into clinically relevant molecular subtypes. Methods Gene copy number, mutation, and expression data of 193 histologically classified oligodendrogliomas from The Cancer Genome Atlas (TCGA) were analyzed by well-established computational approaches (unsupervised clustering, statistical testing, network inference). Results We applied hierarchical clustering to tumor gene copy number profiles and revealed three molecular subgroups within histologically classified oligodendrogliomas. We further screened these subgroups for molecular glioma markers (1p/19q co-deletion, IDH mutation, gain of chromosome 7 and loss of chromosome 10) and found that our subgroups largely resemble known molecular glioma subtypes. We excluded glioblastoma-like tumors (7a10d subgroup) and derived a gene expression signature distinguishing histologically classified oligodendrogliomas with concurrent 1p/19q co-deletion and IDH mutation (1p/19q subgroup) from those with predominant IDH mutation alone (IDHme subgroup). Interestingly, many signature genes were part of signaling pathways involved in the regulation of cell proliferation, differentiation, migration, and cell-cell contacts. We further learned a gene regulatory network associated with the gene expression signature revealing novel putative major regulators with functions in cytoskeleton remodeling (e.g. APBB1IP, VAV1, ARPC1B), apoptosis (CCNL2, CREB3L1), and neural development (e.g. MYTIL, SCRT1, MEF2C) potentially contributing to the manifestation of differences between both subgroups. Moreover, we revealed characteristic expression differences of several HOX and SOX transcription factors suggesting the activity of different glioma stemness programs in both subgroups. Conclusions We show that gene copy number profiles alone are sufficient to derive molecular subgroups of histologically classified oligodendrogliomas that are well-embedded into general glioma classification schemes. Moreover, our revealed novel putative major regulators and characteristic stemness signatures indicate that different developmental programs might be active in these subgroups, providing a basis for future studies.

Background Molecular data of histologically classified oligodendrogliomas are available offering the possibility to stratify these human brain tumors into clinically relevant molecular subtypes. Methods Gene copy number, mutation, and expression data of 193 histologically classified oligodendrogliomas from The Cancer Genome Atlas (TCGA) were analyzed by well-established computational approaches (unsupervised clustering, statistical testing, network inference). Results We applied hierarchical clustering to tumor gene copy number profiles and revealed three molecular subgroups within histologically classified oligodendrogliomas. We further screened these subgroups for molecular glioma markers (1p/19q co-deletion, IDH mutation, gain of chromosome 7 and loss of chromosome 10) and found that our subgroups largely resemble known molecular glioma subtypes. We excluded glioblastoma-like tumors (7a10d subgroup) and derived a gene expression signature distinguishing histologically classified oligodendrogliomas with concurrent 1p/19q co-deletion and IDH mutation (1p/19q subgroup) from those with predominant IDH mutation alone (IDHme subgroup). Interestingly, many signature genes were part of signaling pathways involved in the regulation of cell proliferation, differentiation, migration, and cell-cell contacts. We further learned a gene regulatory network associated with the gene expression signature revealing novel putative major regulators with functions in cytoskeleton remodeling (e.g. APBB1IP, VAV1, ARPC1B), apoptosis (CCNL2, CREB3L1), and neural development (e.g. MYTIL, SCRT1, MEF2C) potentially contributing to the manifestation of differences between both subgroups. Moreover, we revealed characteristic expression differences of several HOX and SOX transcription factors suggesting the activity of different glioma stemness programs in both subgroups. Conclusions We show that gene copy number profiles alone are sufficient to derive molecular subgroups of histologically classified oligodendrogliomas that are well-embedded into general glioma classification schemes. Moreover, our revealed novel putative major regulators and characteristic stemness signatures indicate that different developmental programs might be active in these subgroups, providing a basis for future studies.