Fully inert subgroups of torsion-complete p-groups
Copyright policy )Fully inert subgroups of torsion-complete p-groups
A subgroup \(H\) of an abelian group \(G\) is called fully inert if the quotient \((H+\phi(H))/H\) is finite for every endomorphism \(\phi\) of \(G\). This is a common generalization of the notions of fully invariant, finite and finite-index subgroups. Two subgroups \(K\) and \(L\) of \(G\) are commensurable if \([K:L\cap K]\) and \([L:L\cap K]\) are both finite. \(\phi\)-inert subgroups were a basic tool in the definition of intrinsic algebraic entropy introduced in [\textit{D. Dikranjan} et al., J. Pure Appl. Algebra 219, No. 7, 2933--2961 (2015; Zbl 1155.20041)], which in turn is a variant of the notion of algebraic entropy that was investigated in depth in [\textit{D. Dikranjan} et al., Trans. Am. Math. Soc. 361, No. 7, 3401--3434 (2009; Zbl 1176.20057)]. The authors states that fully inert subgroups of torsion-complete abelian \(p\)-groups are commensurable with fully invariant subgroups. This result is analogues to corresponding statement for direct sums of \(p\)-cyclic groups [\textit{B. Goldsmith} et al., J. Algebra 419, 332--349 (2014; Zbl 1305.20063)].
- University of Padua Italy
- National Institute for Nuclear Physics Italy
- Technological University Dublin Ireland
fully inert subgroup, Algebra and Number Theory, commensurable subgroups, Direct sums, direct products, etc. for abelian groups, Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups, direct sum of cyclic \(p\)-groups, Subgroups of abelian groups, fully invariant subgroup, Torsion groups, primary groups and generalized primary groups, torsion-complete \(p\)-group
fully inert subgroup, Algebra and Number Theory, commensurable subgroups, Direct sums, direct products, etc. for abelian groups, Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups, direct sum of cyclic \(p\)-groups, Subgroups of abelian groups, fully invariant subgroup, Torsion groups, primary groups and generalized primary groups, torsion-complete \(p\)-group
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