We study the 14-dimensional real representation of
the finite simple group G2(3) and relate it
(a) the compact real form of the Lie group G2,
(b) the smallest Ree group and
in characteristic 3.
In particular, we give a set of generators which leads to a new
and easy proof that the group is indeed G2(3).
We give a description of the double cover of Conway's group in terms of right multiplications by 3 × 3 matrices over the octonions. This leads to simple sets of generators for many of the maximal subgroups, including a uniform construction of the Suzuki chain of subgroups.
It is known that for finite simple groups it is possible for a faithful absolutely irreducible module to have 1-cohomology of dimension at least 3. However, until now no explicit examples have been found. We present two explicit examples where the dimension is exactly 3. It remains an open question as to whether the dimension can be bigger than 3.
Friedreich ataxia (FRDA) is an autosomal recessive neurodegenerative and cardiodegenerative disorder caused by decreased expression and/or function of the protein frataxin. Frataxin is involved in iron-sulfur cluster (ISC) biogenesis. Decreased expression of frataxin is associated with decreased ISC biogenesis, mitochondrial iron accumulation, and increased oxidative stress, all of which contribute to mitochondrial dysfunction. Frataxin binds iron and may chaperone iron for ISC biogenesis in the mitochondrial matrix. Frataxin also interacts with multiple proteins in the ISC-biogenesis complex and regulates ISC biogenesis in part through the activation of the cysteine desulfurase. Because decreased frataxin affects cytosolic ISC biogenesis as well as mitochondrial ISC biogenesis, the pathogenesis of FRDA undoubtedly results from decreased activities of both mitochondrial and cytosolic ISC proteins.
In the Kourovka Notebook, Deaconescu asks if |Aut G| ≥ φ(|G|) for all finite groups G, where φ denotes the Euler totient function; and whether G is cyclic whenever |Aut G| = φ(|G|). In an earlier paper we have answered both questions in the negative, and shown that |Aut G|/φ(|G|) can be made arbitrarily small. Here we show that these results remain true if G is restricted to being perfect, or soluble. The problem remains open when G is supersoluble, or nilpotent.
An exploration into the utility of the thermally-induced (metal oxide-mediated) CO2 extrusion and subsequent rearrangement of N-acyl-1,3-oxazolidin-2-ones to form 2-aryl(alkyl)-2-oxazolines is described. The reaction is found to give moderate yields of the corresponding 2-oxazolines. Attempts to employ the above methodology to give enantiopure (R)- or (S)-2,5-diphenyl-2-oxazoline (the latter form being the natural product Oxytriphine) from enantiopure (and crystallographically characterised) (S)-N-benzoyl-5-phenyl-2-oxazolidinone led to the isolation of an essentially racemic product. These protocols are compared to other common methods used to form the oxazoline ring system and are placed into context with previous investigations of such ring forming reactions.