The theory of endomorphism rings of algebraic structures allows, in a natural way, a systematic approach based on the notion of entropy borrowed from dynamical systems. In the present work we introduce a 'dual' notion based upon the replacement of the finite groups used in the definition of algebraic entropy, by subgroups of finite index. The basic properties of this new entropy are established and a connection to Hopfian groups is investigated.
We study the endomorphisms φ of abelian groups G having
a “small” algebraic entropy h (where “small” usually means
h(φ) < log 2). Using essentially elementary tools from linear algebra, we show that this study can be carried out in the group Q^d, where an automorphism φ with h(φ) < log 2 must have all eigenvalues in the open circle of radius 2, centered at 0 and φ must leave invariant a lattice in Q^d, i.e., be essentially an automorphism of Z^d. In particular, all eigenvalues of an automorphism φ with h(φ) = 0 must be roots of unity. This is a particular case of a more general fact known as Algebraic Yuzvinskii Theorem. We discuss other particular cases of this fact and we give some applications of our main results
The classical notions of transitivity and full transitivity in Abelian p-groups have natural extensions to concepts called Krylov and weak transitivity.The interconnections between these four types of transitivity are determined for Abelian p-groups; there is a marked difference in the relationships when the prime p is equal to 2. In the final section the relationship between full and Krylov transitivity is examined in the case of mixed Abelian groups which are p-local in the sense that multiplication by an integer relatively prime to p is an automorphism.
Necessary and sufficient conditions to ensure that the direct sum of two Abelian groups with zero entropy is again of zero entropy are still unknown; interestingly the same problem is also unresolved for direct sums of Hopfian and co-Hopfian groups.We obtain sufficient conditions in some situations by placing restrictions on the homomorphisms between the groups. There are clear similarities between the various cases but there is not a simple duality involved.
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