Almost completely decomposable groups with primary regulator quotients and their endomorphism rings

Let X be a block-rigid almost completely decomposable group of ring type with regulator A and p-primary regulator quotient X/A such that p ^l = exp X/A with natural l > 1. From the well-known fact p ^l End A ⊂ End X ⊂ End A it follows that End X = End X ∪ End A and p ^l End A = End X ∪ p ^l End A. Generalizing these, we determine the chain End X = ɛ _A ^(l) ⊂ ɛ _A ^(l−1) ⊂ ɛ _A ^(l−2) ⊂ ⋯ ⊂ ɛ _A ⁽¹⁾ ⊂ ɛ _A ⁽⁰⁾ = End A, satisfying p ^l−k ɛ _A ^(k) = End X ∪p ^l−k End A, and construct groups X _k ^′ and such that ɛ _A ^(k) = Hom , where k = 1, 2,..., l − 1. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 2, pp. 17–38, 2006.