The symmetric groups $S_n, n\geq 4$, and finite non-abelian simple groups are not embeddable in any Riordan group
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910024834482176 |
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| author | He, Tian-Xiao Krylov, Nikolai A. |
| author_facet | He, Tian-Xiao Krylov, Nikolai A. |
| contents | We prove that the symmetric group of degree greater than three cannot be embedded into the Riordan group with coefficients in any commutative ring. We also prove the impossibility to embed finite non-abelian simple groups. As a closely related topic, we show why all truncated Riordan groups are solvable, in stark contrast to the unsolvability of the infinite-sized Riordan groups. Finally, we give an explicit embedding of the alternating group $A_4$ into the Lagrange subgroup with coefficients in a certain commutative ring, and prove that $A_4$ cannot be embedded into a substitution group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_03056 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The symmetric groups $S_n, n\geq 4$, and finite non-abelian simple groups are not embeddable in any Riordan group He, Tian-Xiao Krylov, Nikolai A. Group Theory Combinatorics 05A05, 20B30, 20F16, 20H25 We prove that the symmetric group of degree greater than three cannot be embedded into the Riordan group with coefficients in any commutative ring. We also prove the impossibility to embed finite non-abelian simple groups. As a closely related topic, we show why all truncated Riordan groups are solvable, in stark contrast to the unsolvability of the infinite-sized Riordan groups. Finally, we give an explicit embedding of the alternating group $A_4$ into the Lagrange subgroup with coefficients in a certain commutative ring, and prove that $A_4$ cannot be embedded into a substitution group. |
| title | The symmetric groups $S_n, n\geq 4$, and finite non-abelian simple groups are not embeddable in any Riordan group |
| topic | Group Theory Combinatorics 05A05, 20B30, 20F16, 20H25 |
| url | https://arxiv.org/abs/2508.03056 |