In this short note we identify the unit component of Newman--Unti (NU) group in fine very strong topology. previous work, has been endowed with an infinite-dimensional Lie structure, while full NU-group does not support such a structure.

The unit conjecture, commonly attributed to Kaplansky, predicts that if $K$ is a field and $G$ torsion-free group, then the only units of group ring $K[G]$ are trivial units, is, non-zero scalar multiples elements. We give concrete counterexample this conjecture; virtually abelian order two.

Let G be a finite group, p prime, and (K,O,F) p-modular system. We prove that the trivial source ring of OG is isomorphic to coherent G-stable tuples (χP), where χP virtual character K[NG(P)/P], P runs through all p-subgroups G, coherence condition equality certain values. use this result describe group orthogonal units as product unit Burnside fusion system with (φP) homomorphisms NG(P)/P→F×. ...