Maximal Subgroups of Compact Lie Groups

This report aims at giving a general overview on the classification of the maximal subgroups of compact Lie groups (not necessarily connected). In the first part, it is shown that these fall naturally into three types: (1) those of trivial type, which are simply defined as inverse images of maximal subgroups of the corresponding component group under the canonical projection and whose classification constitutes a problem in finite group theory, (2) those of normal type, whose connected one-component is a normal subgroup, and (3) those of normalizer type, which are the normalizers of their own connected one-component. It is also shown how to reduce the classification of maximal subgroups of the last two types to: (2) the classification of the finite maximal Sigma-invariant subgroups of centerfree connected compact simple Lie groups and (3) the classification of the Sigma-primitive subalgebras of compact simple Lie algebras, where Sigma is a subgroup of the corresponding outer automorphism group. In the second part, we explicitly compute the normalizers of the primitive subalgebras of the compact classical Lie algebras (in the corresponding classical groups), thus arriving at the complete classification of all (non-discrete) maximal subgroups of the compact classical Lie groups.

EXTENSION OF AUTOMORPHISMS OF SUBGROUPS

Let G be a group such that, for any subgroup H of G, every automorphism of H can be extended to an automorphism of G. Such a group G is said to be of injective type. The finite abelian groups of injective type are precisely the quasi-injective groups. We prove that a finite non-abelian group G of injective type has even order. If, furthermore, G is also quasi-injective, then we prove that G = K x B, with B a quasi-injective abelian group of odd order and either K = Q(8) (the quaternion group of order 8) or K = Dih(A), a dihedral group on a quasi-injective abelian group A of odd order coprime with the order of B. We give a description of the supersoluble finite groups of injective type whose Sylow 2-subgroup are abelian showing that these groups are, in general, not quasi-injective. In particular, the characterisation of such groups is reduced to that of finite 2-groups that are of injective type. We give several restrictions on the latter. We also show that the alternating group A(5) is of injective type but that the binary icosahedral group SL(2, 5) is not.

Profiling Three-Dimensional Nuclear Telomeric Architecture of Myelodysplastic Syndromes and Acute Myeloid Leukemia Defines Patient Subgroups

Purpose: Myelodysplastic syndromes (MDS) are a group of disorders characterized by cytopenias, with a propensity for evolution into acute myeloid leukemias (AML). This transformation is driven by genomic instability, but mechanisms remain unknown. Telomere dysfunction might generate genomic instability leading to cytopenias and disease progression. Experimental Design: We undertook a pilot study of 94 patients with MDS (56 patients) and AML (38 patients). The MDS cohort consisted of refractory cytopenia with multilineage dysplasia (32 cases), refractory anemia (12 cases), refractory anemia with excess of blasts (RAEB) 1 (8 cases), RAEB2 (1 case), refractory anemia with ring sideroblasts (2 cases), and MDS with isolated del(5q) (1 case). The AML cohort was composed of AML-M4 (12 cases), AML-M2 (10 cases), AML-M5 (5 cases), AML-M0 (5 cases), AML-M1 (2 cases), AML-M4eo (1 case), and AML with multidysplasia-related changes (1 case). Three-dimensional quantitative FISH of telomeres was carried out on nuclei from bone marrow samples and analyzed using TeloView. Results: We defined three-dimensional nuclear telomeric profiles on the basis of telomere numbers, telomeric aggregates, telomere signal intensities, nuclear volumes, and nuclear telomere distribution. Using these parameters, we blindly subdivided the MDS patients into nine subgroups and the AML patients into six subgroups. Each of the parameters showed significant differences between MDS and AML. Combining all parameters revealed significant differences between all subgroups. Three-dimensional telomeric profiles are linked to the evolution of telomere dysfunction, defining a model of progression from MDS to AML. Conclusions: Our results show distinct three-dimensional telomeric profiles specific to patients with MDS and AML that help subgroup patients based on the severity of telomere dysfunction highlighted in the profiles. Clin Cancer Res; 18(12); 3293-304. (C) 2012 AACR.