# Difference between revisions of "Element structure of special linear group:SL(2,5)"

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This article gives detailed information about the element structure of special linear group:SL(2,5), which is a group of order 120.

## Conjugacy class structure

### Relationship with conjugacy class structure for an arbitrary special linear group of degree two

Further information: element structure of special linear group of degree two

Nature of conjugacy class Eigenvalue pairs of all conjugacy classes Characteristic polynomials of all conjugacy classes Minimal polynomials of all conjugacy classes Size of conjugacy class Number of such conjugacy classes Total number of elements Semisimple? Diagonalizable over $\mathbb{F}_q$? Splits in $SL_2$ relative to $GL_2$? Representative matrices (one per conjugacy class)
Scalar $\{ 1, 1 \}$ or $\{ -1,-1\}$ $x^2 - 2x + 1$ or $x^2 + 2x + 1$ $x - 1$ or $x + 1$ 1 2 2 Yes Yes No $\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}$ and $\begin{pmatrix} -1 & 0 \\ 0 & -1\\\end{pmatrix}$
Not diagonal, Jordan block of size two $\{ 1, 1 \}$ or $\{ -1,-1\}$ $x^2 - 2x + 1$ or $x^2 + 2x + 1$ $x^2 - 2x + 1$ or $x^2 + 2x + 1$ 12 4 48 No No Yes $\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}$, $\begin{pmatrix} 1 & 2 \\ 0 & 1 \\\end{pmatrix}$, $\begin{pmatrix} -1 & 1 \\ 0 & -1 \\\end{pmatrix}$, $\begin{pmatrix} -1 & 2 \\ 0 & -1 \\\end{pmatrix}$
Diagonalizable over field:F25, not over field:F5. Must necessarily have no repeated eigenvalues. pair of square roots of $2$ in field:F25, pair of square roots of $3$ in field:F25 $x^2 - x + 1$, $x^2 + x + 1$ $x^2 - x + 1$, $x^2 + x +1$ 20 2 40 Yes No No $\begin{pmatrix}0 & -1\\ 1 & 1\\\end{pmatrix}$, $\begin{pmatrix}0 & -1\\ 1 & -1\\\end{pmatrix}$
Diagonalizable over field:F5 with distinct diagonal entries $2,3$ $x^2 + 1$ $x^2 + 1$ 30 1 30 Yes Yes No $\begin{pmatrix} 2 & 0 \\ 0 & 3 \\\end{pmatrix}$
Total NA NA NA NA 9 120 72 32 48 NA

Nature of conjugacy class Eigenvalue pairs of all conjugacy classes Characteristic polynomials of all conjugacy classes Minimal polynomials of all conjugacy classes Size of conjugacy class (generic odd $q$) Size of conjugacy class ($q = 5$) Number of such conjugacy classes (generic odd $q$) Number of such conjugacy classes ($q = 5$) Total number of elements (generic odd $q$) Total number of elements ($q = 5$) Representative matrices (one per conjugacy class)
Scalar $\{ 1, 1 \}$ or $\{ -1,-1\}$ $x^2 - 2x + 1$ or $x^2 + 2x + 1$ $x - 1$ or $x + 1$ 1 1 2 2 2 2 $\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}$ and $\begin{pmatrix} -1 & 0 \\ 0 & -1\\\end{pmatrix}$
Not diagonal, Jordan block of size two $\{ 1, 1 \}$ or $\{ -1,-1\}$ $x^2 - 2x + 1$ or $x^2 + 2x + 1$ $x^2 - 2x + 1$ or $x^2 + 2x + 1$ $(q^2 - 1)/2$ 12 4 4 $2(q^2 - 1)$ 48 [SHOW MORE]
Diagonalizable over field:F25, not over field:F5. Must necessarily have no repeated eigenvalues. $\{ 2 + \sqrt{3}, 2 - \sqrt{3} \}$ and $\{ -2 + \sqrt{3}, -2 - \sqrt{3} \}$, where $\sqrt{3}$ is interpreted a an element of field:F25 that squares to 3 $x^2 - x + 1$, $x^2 + x + 1$ $x^2 - x + 1$, $x^2 + x + 1$ $q(q - 1)$ 20 $(q - 1)/2$ 2 $q(q - 1)^2/2$ 40 $\begin{pmatrix}0 & -1\\ 1 & 1\\\end{pmatrix}$, $\begin{pmatrix}0 & -1\\ 1 & -1\\\end{pmatrix}$
Diagonalizable over field:F5 with distinct diagonal entries $\{ 2,3 \}$ $x^2 + 1$ $x^2 + 1$ $q(q+1)$ 30 $(q - 3)/2$ 1 $q(q+1)(q-3)/2$ 30 $\begin{pmatrix} 2 & 0 \\ 0 & 3 \\\end{pmatrix}$
Total NA NA NA NA NA $q + 4$ 9 $q^3 - q$ 120 NA