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# Direct Decompositions of Block-Rigid crq-Groups

DOI link for Direct Decompositions of Block-Rigid crq-Groups

Direct Decompositions of Block-Rigid crq-Groups book

# Direct Decompositions of Block-Rigid crq-Groups

DOI link for Direct Decompositions of Block-Rigid crq-Groups

Direct Decompositions of Block-Rigid crq-Groups book

## ABSTRACT

T heorem 13.1.1. (P rod u ct Form ula) Let X be a block-rigid crq-group with regulator A. Suppose X = X \ 0 X 2 is a direct decomposition of X . Then each X{

is a block-rigid crq-group with regulator A{ = Χ{Γ\ A, and

is a decomposition of the cyclic group X /A . The groups X i/A i are cyclic and have relatively prime orders. Furthermore,

P roof. Since A = R(AT) is fully invariant in X , we obtain the decomposition

It follows that

and since X /A is cyclic, the orders of the cyclic summands X i/A i are pairwise relatively prime. The rest is straightforward using Lemma 12.6.3 and the definition of μτ. □

Theorem 13.1.2. (E x isten ce T heorem ) Given a finite antichain T of types, and a set {μρ : p G T} of natural numbers > 1, there exists a crq-group X with Tcr(A) C T and μτ(Χ) = μτ if and only if the following two adm issib ility con ditions are satisfied.