Element structure of general linear group of degree three over a finite field
This article gives specific information, namely, element structure, about a family of groups, namely: general linear group of degree three.
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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.
|Group||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 with equal diagonal entries, hence a scalar||where||1||Yes||Yes|
|Diagonalizable over with one eigenvalue having multiplicity two, the other eigenvalue having multiplicity one||where , both in||Yes||Yes|
|Diagonalizable over with all distinct diagonal entries||, all distinct elements of||same as characteristic polynomial||Yes||Yes|
|Diagonalizable over , not over||Distinct Galois conjugate triple of elements in . If one of the elements is , the other two are and .||irreducible degree three polynomial over||same as characteristic polynomial||Yes||No|
|One eigenvalue is in , the other two are in||one element of , pair of Galois conjugates over in .||product of linear polynomial and irreducible degree two polynomial over||same as characteristic polynomial||Yes||No|
|Has Jordan blocks of sizes 2 and 1 with distinct eigenvalues over||with ,||same as characteristic polynomial||No||No|
|Has Jordan blocks of sizes 2 and 1 with equal eigenvalues over||with||No||No|
|Has Jordan block of size 3||with||same as characteristic polynomial||No||No|