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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.16650 |
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| _version_ | 1866911393680195584 |
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| author | Harper, Scott Quick, Martyn |
| author_facet | Harper, Scott Quick, Martyn |
| contents | Famously, every finite simple group $G$ can be generated by a pair of elements. Moreover, Liebeck and Shalev (1995) proved that the probability that a pair of elements generate $G$ tends to $1$ as $|G| \to \infty$. More generally, work of Lucchini and Menegazzo (1997) implies that $G$ can be generated by a pair of elements whenever $G$ has a unique chief series. In this paper, we generalize the theorem of Liebeck and Shalev by proving that if $G$ has a unique chief series and the unique simple quotient of $G$ is $S$, then the probability that a pair of elements generate $G$ tends to $1$ as $|S| \to \infty$. As a consequence of our main theorem, for any profinite group $G$ where the open normal subgroups form a chain, the probability that a pair of elements topologically generate $G$ is positive. Along the way, we establish results on the maximal subgroup zeta function of groups with a unique minimal normal subgroup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16650 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The probability of generating a uniserial group Harper, Scott Quick, Martyn Group Theory Famously, every finite simple group $G$ can be generated by a pair of elements. Moreover, Liebeck and Shalev (1995) proved that the probability that a pair of elements generate $G$ tends to $1$ as $|G| \to \infty$. More generally, work of Lucchini and Menegazzo (1997) implies that $G$ can be generated by a pair of elements whenever $G$ has a unique chief series. In this paper, we generalize the theorem of Liebeck and Shalev by proving that if $G$ has a unique chief series and the unique simple quotient of $G$ is $S$, then the probability that a pair of elements generate $G$ tends to $1$ as $|S| \to \infty$. As a consequence of our main theorem, for any profinite group $G$ where the open normal subgroups form a chain, the probability that a pair of elements topologically generate $G$ is positive. Along the way, we establish results on the maximal subgroup zeta function of groups with a unique minimal normal subgroup. |
| title | The probability of generating a uniserial group |
| topic | Group Theory |
| url | https://arxiv.org/abs/2601.16650 |