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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2402.00968 |
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| _version_ | 1866916112639197184 |
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| author | Vyshnevetskiy, Oleksandr |
| author_facet | Vyshnevetskiy, Oleksandr |
| contents | Let $P$ be a probability on a finite group $G$, ${P^{(n)}}$ $n$-fold convolution of $P$ on $G$. Under mild condition, ${P^{(n)}}$ at $n \to \infty $ converges to the uniform probability on the group $G$. If $A = \left\{ {g \in G,\;P\left( g \right) \ne 0} \right\}$ be the carrier of the probability $P$, then ${A^n} = \left\{ {{a_1} \cdot ... \cdot {a_n},\;\;{a_1},...,{a_n} \in A} \right\}$ be the carrier of probability ${P^{(n)}}$. One of necessary and sufficient conditions for the mentioned convergence is: sequence ${A^n}$ at $n \to \infty $ stabilizes on $G$, i.e. ${A^k}$ = ${A^{k + 1}} = ... = G$ for a natural number $k$. In other words, product of some multipliers equal to $A$ is $G$. The carrier $A$ is in general case any nonempty subset of group $G$. In the paper we find a condition under which product of some subsets of $G$ is $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_00968 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Products of subsets of group that equal the group Vyshnevetskiy, Oleksandr Group Theory Probability 20D99 (Primary) 60B15 (Secondary) Let $P$ be a probability on a finite group $G$, ${P^{(n)}}$ $n$-fold convolution of $P$ on $G$. Under mild condition, ${P^{(n)}}$ at $n \to \infty $ converges to the uniform probability on the group $G$. If $A = \left\{ {g \in G,\;P\left( g \right) \ne 0} \right\}$ be the carrier of the probability $P$, then ${A^n} = \left\{ {{a_1} \cdot ... \cdot {a_n},\;\;{a_1},...,{a_n} \in A} \right\}$ be the carrier of probability ${P^{(n)}}$. One of necessary and sufficient conditions for the mentioned convergence is: sequence ${A^n}$ at $n \to \infty $ stabilizes on $G$, i.e. ${A^k}$ = ${A^{k + 1}} = ... = G$ for a natural number $k$. In other words, product of some multipliers equal to $A$ is $G$. The carrier $A$ is in general case any nonempty subset of group $G$. In the paper we find a condition under which product of some subsets of $G$ is $G$. |
| title | Products of subsets of group that equal the group |
| topic | Group Theory Probability 20D99 (Primary) 60B15 (Secondary) |
| url | https://arxiv.org/abs/2402.00968 |