Element structure of projective general linear group:PGL(2,9)
This article gives specific information, namely, element structure, about a particular group, namely: projective general linear group:PGL(2,9).
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Conjugacy class structure
There is a total of conjugacy classes.
|Nature of conjugacy class upstairs in||Eigenvalues||Characteristic polynomial||Minimal polynomial||Size of conjugacy class||Number of such conjugacy classes||Total number of elements|
|Diagonalizable over field:F9 with equal diagonal entries, hence a scalar||where||1||1||1|
|Diagonalizable over , not over , eigenvalues are negatives of each other.||Pair of mutually negative conjugate elements of . All such pairs identified.||, a nonzero non-square||Same as characteristic polynomial||36||1|
|Diagonalizable over with mutually negative diagonal entries.||, all such pairs identified.||, all identified||Same as characteristic polynomial||45||1||45|
|Diagonalizable over , not over , eigenvalues are not negatives of each other.||Pair of conjugate elements of . Each pair identified with anything obtained by multiplying both elements of it by an element of .||, , irreducible; with identification.||Same as characteristic polynomial||72||4||288|
|Not diagonal, has Jordan block of size two||(multiplicity 2). Each conjugacy class has one representative of each type.||Same as characteristic polynomial||80||1||80|
|Diagonalizable over with distinct diagonal entries whose sum is not zero.||where and . The pairs and are identified.||, again with identification.||Same as characteristic polynomial.||90||3||270|