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| Format: | Preprint |
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2018
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| Online Access: | https://arxiv.org/abs/1802.10471 |
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| _version_ | 1866908958156914688 |
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| author | Biswas, Arindam |
| author_facet | Biswas, Arindam |
| contents | We introduce a notion of finite approximate subloops in Moufang loops, with emphasis on the commutative case. For arbitrary Moufang loops we establish intrinsic product-set identities and covering consequences without passing through associative quotients and obtain a finite-kernel reduction principle: approximate-subloop structure descends through homomorphisms onto groups with finite kernel, and inverse results in the quotient lift back to the loop. In particular, this yields a complete reduction in the two-generated case. For commutative Moufang loops, using their local finite-by-abelian structure, we deduce a Freiman-type theorem showing that a finite approximate subloop is contained in the pullback of a coset progression from a suitable local abelian quotient, with quantitative bounds depending only on the corresponding finite kernel. We then obtain a uniform version for approximate subloops generating an $m$-generated subloop. When the local abelian quotient has bounded torsion, we get a polynomial covering theorem by cosets of a finite subloop, deduced from the bounded-torsion polynomial Freiman--Ruzsa theorem in the abelian quotient; in particular, this applies to commutative Moufang loops of exponent $3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1802_10471 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Approximate Subloops in Moufang Loops Biswas, Arindam Group Theory 20N05, 20F69, 11B30, 05B15 We introduce a notion of finite approximate subloops in Moufang loops, with emphasis on the commutative case. For arbitrary Moufang loops we establish intrinsic product-set identities and covering consequences without passing through associative quotients and obtain a finite-kernel reduction principle: approximate-subloop structure descends through homomorphisms onto groups with finite kernel, and inverse results in the quotient lift back to the loop. In particular, this yields a complete reduction in the two-generated case. For commutative Moufang loops, using their local finite-by-abelian structure, we deduce a Freiman-type theorem showing that a finite approximate subloop is contained in the pullback of a coset progression from a suitable local abelian quotient, with quantitative bounds depending only on the corresponding finite kernel. We then obtain a uniform version for approximate subloops generating an $m$-generated subloop. When the local abelian quotient has bounded torsion, we get a polynomial covering theorem by cosets of a finite subloop, deduced from the bounded-torsion polynomial Freiman--Ruzsa theorem in the abelian quotient; in particular, this applies to commutative Moufang loops of exponent $3$. |
| title | Approximate Subloops in Moufang Loops |
| topic | Group Theory 20N05, 20F69, 11B30, 05B15 |
| url | https://arxiv.org/abs/1802.10471 |