On pre-Lie rings related to some non-Lazard braces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916665430638592 |
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| author | Smoktunowicz, Agata |
| author_facet | Smoktunowicz, Agata |
| contents | Let A be a brace of cardinality $p^{n}$ for some prime number $p$. Suppose that either (i) the additive group of brace $A$ has rank smaller than $p-3$, or (ii) $A^{\frac {p-1}2}\subseteq pA$ or (iii) $p^{i}A$ is an ideal in in $A$ for each $i$.
It is shown that there is a pre-Lie ring associated to brace $A$. The left nilpotency index of this pre-Lie ring can be arbitrarily large.
Let $A$ be a brace of cardinality $p^{n}$ for some prime number $p$. Denote $ann(p^{i})=\{a\in A: p^{i}a=0\}$.
Suppose that for $i=1,2,\ldots $ and all $a,b\in A$ we have
\[a*(a*(\cdots *a*b))\in pA, a*(a*(\cdots *a*ann(p^{i})))\in ann(p^{i-1})\] where $a$ appears less than $\frac {p-1}4$ times in this expression.
Let $k$ be such that $p^{k(p-1)}A=0$. It is shown that the brace $A/ann(p^{4k})$ is obtained from a left nilpotent pre-Lie ring by a formula which depends only on the additive group of brace $A$. We also obtain some applications of this result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_05924 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On pre-Lie rings related to some non-Lazard braces Smoktunowicz, Agata Group Theory 17D90, 20F18, 20F40, 17B70, 20D15 Let A be a brace of cardinality $p^{n}$ for some prime number $p$. Suppose that either (i) the additive group of brace $A$ has rank smaller than $p-3$, or (ii) $A^{\frac {p-1}2}\subseteq pA$ or (iii) $p^{i}A$ is an ideal in in $A$ for each $i$. It is shown that there is a pre-Lie ring associated to brace $A$. The left nilpotency index of this pre-Lie ring can be arbitrarily large. Let $A$ be a brace of cardinality $p^{n}$ for some prime number $p$. Denote $ann(p^{i})=\{a\in A: p^{i}a=0\}$. Suppose that for $i=1,2,\ldots $ and all $a,b\in A$ we have \[a*(a*(\cdots *a*b))\in pA, a*(a*(\cdots *a*ann(p^{i})))\in ann(p^{i-1})\] where $a$ appears less than $\frac {p-1}4$ times in this expression. Let $k$ be such that $p^{k(p-1)}A=0$. It is shown that the brace $A/ann(p^{4k})$ is obtained from a left nilpotent pre-Lie ring by a formula which depends only on the additive group of brace $A$. We also obtain some applications of this result. |
| title | On pre-Lie rings related to some non-Lazard braces |
| topic | Group Theory 17D90, 20F18, 20F40, 17B70, 20D15 |
| url | https://arxiv.org/abs/2410.05924 |