The maximal size of a minimal generating set
A generating set for a finite group G is minimal if no proper subset generates G, and $m(G)$ denotes the maximal size of a minimal generating set for G. We prove a conjecture of Lucchini, Moscatiello and Spiga by showing that there exist $a,b> 0$ such that any finite group G satisfi...
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2023-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509423000713/type/journal_article |
| _version_ | 1797745888307707904 |
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| author | Scott Harper |
| author_facet | Scott Harper |
| author_sort | Scott Harper |
| collection | DOAJ |
| description | A generating set for a finite group G is minimal if no proper subset generates G, and
$m(G)$
denotes the maximal size of a minimal generating set for G. We prove a conjecture of Lucchini, Moscatiello and Spiga by showing that there exist
$a,b> 0$
such that any finite group G satisfies
$m(G) \leqslant a \cdot \delta (G)^b$
, for
$\delta (G) = \sum _{p \text { prime}} m(G_p)$
, where
$G_p$
is a Sylow p-subgroup of G. To do this, we first bound
$m(G)$
for all almost simple groups of Lie type (until now, no nontrivial bounds were known except for groups of rank
$1$
or
$2$
). In particular, we prove that there exist
$a,b> 0$
such that any finite simple group G of Lie type of rank r over the field
$\mathbb {F}_{p^f}$
satisfies
$r + \omega (f) \leqslant m(G) \leqslant a(r + \omega (f))^b$
, where
$\omega (f)$
denotes the number of distinct prime divisors of f. In the process, we confirm a conjecture of Gill and Liebeck that there exist
$a,b> 0$
such that a minimal base for a faithful primitive action of an almost simple group of Lie type of rank r over
$\mathbb {F}_{p^f}$
has size at most
$ar^b + \omega (f)$
. |
| first_indexed | 2024-03-12T15:29:24Z |
| format | Article |
| id | doaj.art-46d5df7c84af44008eff16cb3bd0e7b2 |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-03-12T15:29:24Z |
| publishDate | 2023-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-46d5df7c84af44008eff16cb3bd0e7b22023-08-10T08:17:56ZengCambridge University PressForum of Mathematics, Sigma2050-50942023-01-011110.1017/fms.2023.71The maximal size of a minimal generating setScott Harper0https://orcid.org/0000-0002-0056-2914School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS, United Kingdom; E-mail:A generating set for a finite group G is minimal if no proper subset generates G, and $m(G)$ denotes the maximal size of a minimal generating set for G. We prove a conjecture of Lucchini, Moscatiello and Spiga by showing that there exist $a,b> 0$ such that any finite group G satisfies $m(G) \leqslant a \cdot \delta (G)^b$ , for $\delta (G) = \sum _{p \text { prime}} m(G_p)$ , where $G_p$ is a Sylow p-subgroup of G. To do this, we first bound $m(G)$ for all almost simple groups of Lie type (until now, no nontrivial bounds were known except for groups of rank $1$ or $2$ ). In particular, we prove that there exist $a,b> 0$ such that any finite simple group G of Lie type of rank r over the field $\mathbb {F}_{p^f}$ satisfies $r + \omega (f) \leqslant m(G) \leqslant a(r + \omega (f))^b$ , where $\omega (f)$ denotes the number of distinct prime divisors of f. In the process, we confirm a conjecture of Gill and Liebeck that there exist $a,b> 0$ such that a minimal base for a faithful primitive action of an almost simple group of Lie type of rank r over $\mathbb {F}_{p^f}$ has size at most $ar^b + \omega (f)$ .https://www.cambridge.org/core/product/identifier/S2050509423000713/type/journal_article20F0520B0520E2820E32 |
| spellingShingle | Scott Harper The maximal size of a minimal generating set Forum of Mathematics, Sigma 20F05 20B05 20E28 20E32 |
| title | The maximal size of a minimal generating set |
| title_full | The maximal size of a minimal generating set |
| title_fullStr | The maximal size of a minimal generating set |
| title_full_unstemmed | The maximal size of a minimal generating set |
| title_short | The maximal size of a minimal generating set |
| title_sort | maximal size of a minimal generating set |
| topic | 20F05 20B05 20E28 20E32 |
| url | https://www.cambridge.org/core/product/identifier/S2050509423000713/type/journal_article |
| work_keys_str_mv | AT scottharper themaximalsizeofaminimalgeneratingset AT scottharper maximalsizeofaminimalgeneratingset |