The class number \(k(G)\) of a group \(G\) is the number of conjugacy classes of \(G\). In 1903, Landau proved in [Lan03] that for every \(n \in \mathbb{N}\), there are only finitely many finite groups with exactly \(n\) conjugacy classes. Thus, it is possible to classify finite groups according to their class number.
The SmallClassNr package provides access to the finite groups with class number at most \(20\). These groups were classified in the following papers:
\(k(G) \leq 5\), by Miller in [Mil11] and independently by Burnside in [Bur11]
\(k(G) = 6,7\), by Poland in [Pol68]
\(k(G) = 8\), by Kosvintsev in [Kos74]
\(k(G) = 9\), by Odincov and Starostin in [OS76]
\(k(G) = 10,11\), by Vera López and Vera López in [VV85]
\(k(G) = 12\), by Vera López and Vera López in [VV86]
\(k(G) = 13, 14\), by Vera López and Sangroniz in [VS07]
\(k(G) = 15, \dots, 20\), by Breuer in [Bre26]
Remarks:
In [VV85], three distinct groups of the form \((C_5 \times C_5) \rtimes C_4\) with class number \(10\) are given. However, only two such groups exist, being the ones with IdClassNr equal to [10,25] and [10,26].
In [VV86], 48 groups with class number 12 are listed. There are actually 51 such groups, the three groups missing in [VV86] are provided in the appendix of [VS07]. These are the groups with IdClassNr equal to [12,13], [12,16] and [12,39].
The classification of groups with 15 to 20 conjugacy classes has not been refereed yet, and is therefore disabled by default. It can still be accessed via the command below.
‣ LoadUnverifiedSmallClassNrGroups( ) | ( function ) |
Loads the groups with class numbers 15 to 20.
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