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Classifying finite monomial linear groups of prime degree in characteristic zero. (English) Zbl 07456360

Summary: Let \(p\) be a prime and let \(\mathbb{C}\) be the complex field. We explicitly classify the finite solvable irreducible monomial subgroups of \(\mathrm{GL}(p,\mathbb{C})\) up to conjugacy. That is, we give a complete and irredundant list of \(\mathrm{GL}(p,\mathbb{C})\)-conjugacy class representatives as generating sets of monomial matrices. Copious structural information about non-solvable finite irreducible monomial subgroups of \(\mathrm{GL}(p,\mathbb{C})\) is also proved, enabling a classification of all such groups bar one family. We explain the obstacles in that exceptional case. For \(p\leq 3\), we classify all finite irreducible subgroups of \(\mathrm{GL}(p,\mathbb{C})\). Our classifications are available publicly in MAGMA.

MSC:

20H20 Other matrix groups over fields
20E99 Structure and classification of infinite or finite groups
20-04 Software, source code, etc. for problems pertaining to group theory

Software:

Magma; GitHub
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Full Text: DOI arXiv

References:

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