Linear representations of soluble groups of finite Morley rank. (English) Zbl 1228.20028
The paper addresses the question of the linearity of connected soluble groups of finite Morley rank. More precisely, sufficient conditions are given for such a group with a nontrivial torsion-free normal nilpotent subgroup to have linear representations with small kernels. The main theorem states that if \(G\) has no infinite subgroup of bounded exponent, then it has a linear representation with kernel contained in \(Z_3(G)\), the third term of the upper central series of \(G\). The proofs use the Maltsev correspondence for torsion-free nilpotent groups and introduce a definable version of the notion of weight space in representation theory. Along the way, it is proved that certain connected soluble groups can be embedded in such groups whose Fitting subgroups have Abelian supplements.
Reviewer: Eric Jaligot (Grenoble)
MSC:
| 20F11 | Groups of finite Morley rank |
| 03C60 | Model-theoretic algebra |
| 20F16 | Solvable groups, supersolvable groups |
| 20C15 | Ordinary representations and characters |
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