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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2410.11098 |
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| _version_ | 1866929629659398144 |
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| author | Bunina, Elena |
| author_facet | Bunina, Elena |
| contents | In this paper, we prove that the endomorphism rings End A and End A' of periodic infinite Abelian groups A and A' are elementarily equivalent if and only if the endomorphism rings of their p-components are elementarily equivalent for all primes p. Additionally, we show that the automorphism groups Aut A and Aut A' of periodic Abelian groups A and A' that do not have 2-components and do not contain cocyclic p-components are elementarily equivalent if and only if, for any prime p, the corresponding p-components A_p and A_p' of A and A' are equivalent in second-order logic if they are not reduced, and are equivalent in second-order logic bounded by the cardinalities of their basic subgroups if they are reduced. For such groups A and A', their automorphism groups are elementarily equivalent if and only if their endomorphism rings are elementarily equivalent, and the automorphism groups of the corresponding p-components for all primes p are elementarily equivalent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11098 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Elementary equivalence of endomorphism rings and automorphism groups of periodic Abelian groups Bunina, Elena Group Theory Logic 20K30, 03C68, 03C85 In this paper, we prove that the endomorphism rings End A and End A' of periodic infinite Abelian groups A and A' are elementarily equivalent if and only if the endomorphism rings of their p-components are elementarily equivalent for all primes p. Additionally, we show that the automorphism groups Aut A and Aut A' of periodic Abelian groups A and A' that do not have 2-components and do not contain cocyclic p-components are elementarily equivalent if and only if, for any prime p, the corresponding p-components A_p and A_p' of A and A' are equivalent in second-order logic if they are not reduced, and are equivalent in second-order logic bounded by the cardinalities of their basic subgroups if they are reduced. For such groups A and A', their automorphism groups are elementarily equivalent if and only if their endomorphism rings are elementarily equivalent, and the automorphism groups of the corresponding p-components for all primes p are elementarily equivalent. |
| title | Elementary equivalence of endomorphism rings and automorphism groups of periodic Abelian groups |
| topic | Group Theory Logic 20K30, 03C68, 03C85 |
| url | https://arxiv.org/abs/2410.11098 |