Retracts of group of unitriangular matrices over the ring
Keywords:
retract of group, unitriangular group of matrices over the ring, conventional retract, transvectional retract, essentially conventional retract, computable groupAbstract
Retract of G group is its H subgroup for which endomorphism φ : G → H is in existence identical to H. This notion comes from topology. Description of retracts for important group grades is an up-to-date target. Direct factors in Abelian groups deflate all plenty of retracts. In the general case the problem of retracts description is far more complicated. It is proved that in the category of finitely generated nilpotent groups of nilpotency step two, sub-group prescribed by finite collection of generating elements is algorithmically unresolvable. In this work we obtained necessary and sufficient condition to be retract for Abelian sub-group of arbitrary dimensionality unitriangular matrices group above ring of integers. It was also proved that any retract of unitriangular matrices group of dimensionality three above ring is isomorphic to its additive group which is locally cyclic. This yields the conclusion that existing algorithm determines whether the given sub-group is a retract or not according to any sub-group of dimensionality three unitriangular matrices group. It is also established that the algorithm is available which determines whether the given retract is transvectional or essentially conventional according to any retract of such a group. Calculability of any retract of solvable group was proved that is torsion-free of finite dimension, and its commutant isolation coincide with commutant. In particular it follows that any retract of group of all triangular matrices group of any finite size with positive diagonal elements above the field of rational numbers is calculable.
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