Element structure of general linear group of degree three over a finite field
From Groupprops
This article gives specific information, namely, element structure, about a family of groups, namely: general linear group of degree three.
View element structure of group families | View other specific information about general linear group of degree three
This article discusses the element structure of the general linear group of degree three over a finite field. The group is where
is the order (size) of the field. We denote by
the prime number that is the characteristic of the field.
Particular cases
Group | ![]() |
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Order of the group | Number of conjugacy classes | Element structure page |
---|---|---|---|---|---|
projective special linear group:PSL(3,2) | 2 | 2 | 168 | 6 | element structure of projective special linear group:PSL(3,2) |
general linear group:GL(3,3) | 3 | 3 | 11232 | 24 | element structure of general linear group:GL(3,3) |
general linear group:GL(3,5) | 5 | 5 | 1488000 | 120 | element structure of general linear group:GL(3,5) |
Conjugacy class structure
There is a total of elements, and a total of
conjugacy classes.
Nature of conjugacy class | Eigenvalues | Characteristic polynomial | Minimal polynomial | Size of conjugacy class | Number of such conjugacy classes | Total number of elements | Semisimple? | Diagonalizable over ![]() |
---|---|---|---|---|---|---|---|---|
Diagonalizable over ![]() |
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1 | ![]() |
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Yes | Yes |
Diagonalizable over ![]() |
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Yes | Yes |
Diagonalizable over ![]() |
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same as characteristic polynomial | ![]() |
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Yes | Yes |
Diagonalizable over ![]() ![]() |
Distinct Galois conjugate triple of elements in ![]() ![]() ![]() ![]() |
irreducible degree three polynomial over ![]() |
same as characteristic polynomial | ![]() |
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Yes | No |
One eigenvalue is in ![]() ![]() |
one element of ![]() ![]() ![]() |
product of linear polynomial and irreducible degree two polynomial over ![]() |
same as characteristic polynomial | ![]() |
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Yes | No |
Has Jordan blocks of sizes 2 and 1 with distinct eigenvalues over ![]() |
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same as characteristic polynomial | ![]() |
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No | No |
Has Jordan blocks of sizes 2 and 1 with equal eigenvalues over ![]() |
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No | No |
Has Jordan block of size 3 | ![]() ![]() |
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same as characteristic polynomial | ![]() |
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No | No |
Total | NA | NA | NA | NA | ![]() |
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NA | NA |