Central subgroup

Abstract: Let G be a compact Lie group and p a fixed prime number. Recall that an elementary abelian p-group is an abelian group isomorphic to (Z/p)r for some r. Jackowski and McClure showed in [10] how to decompose the classifying space BG at the prime p as a homotopy colimit of spaces of the form BC G (V). where V is a nontrivial elementary abelian p-subgroup of G and C G (V) is the centralizer of V in G (see §2). If the center of G is trivial then each of the centralizers C G (V) is a proper subgroup of G, and so in this case the decomposition theorem gives an explicit way of gluing together BG, at least at p, from the classifying spaces of smaller groups. In this paper we will use this decomposition to give parallel inductive proofs of three theorems about BG; the first two theorems are already known but the third is probably new.

Abstract: Four $\ZZ_+$-bigraded complexes with the action of the exceptional infinite-dimensional Lie superalgebra E(3,6) are constructed. We show that all the images and cokernels and all but three kernels of the differentials are irreducible E(3,6)-modules. This is based on the list of singular vectors and the calculation of homology of these complexes. As a result, we obtain an explicit construction of all degenerate irreducible E(3,6)-modules and compute their characters and sizes. Since the group of symmetries of the Standard Model $SU(3) \times SU(2) \times U(1)$ (divided by a central subgroup of order six) is a maximal compact subgroup of the group of automorphisms of E(3,6), our results may have applications to particle physics.

Abstract: In this paper we give necessary and sufficient conditions for the unit group OIl7LG, of the integral group ring TLGof a group G, to be an FC-group. A group is called an FC-group if all its conjugacy classes are finite. Our second result proves that if K is a field of characteristic 0 then OIl KG is an FC-group if and only if T(G), the set of torsion elements of G, is a finite central subgroup of G. In [7] the groups G with OIl7LGnilpotent are characterised and related results mentioned. We state the present results. (1.1) Theorem. OIl7LGis an FC-group if and only if G torsion subgroup T satisfies one of the following: (1.2) T is central in G, (1.3) Tis abelian non-central andfor xEG is an F C-group and its

Abstract: We classify finite Moufang loops which are centrally nilpotent of class 2 in terms of certain cubic forms, concentrating on small Frattini Moufang loops, or SFMLs, which are Moufang loops L with a central subgroup Z of order p such that L/Z is an elementary abelian p-group. (For example, SFM 2-loops are precisely the class of code loops, in the sense of Griess.)More specifically, we first show that the nuclearly-derived subloop (normal associator subloop) of a Moufang loop of class 2 has exponent dividing 6. It follows that the subloop of elements of p-power order is associative for p > 3. Next, we show that if L is an SFML, then L/Z has the structure of a vector space with a symplectic cubic form. We then show that every symplectic cubic form is realized by some SFML and that two SFMLs are isomorphic in a manner preserving the central subgroup Z if and only if their symplectic cubic spaces are isomorphic up to scalar multiple. Consequently, we also obtain an explicit characterization of isotopy in SFM 3-loops. Finally, we extend many of our results to all finite Moufang loops of class 2.