4. Huber's paper revisited : the group

Most of the knowledge about $\mathrm{GF}\! \left( q \right)$ is embedded in the following bijections from $\mathrm{GF}\! \left( q \right)$ onto itself :

$\displaystyle \psi \, :\, x\mapsto x^{p}$	$\displaystyle \mathrm{Frobenius}$
$\displaystyle \overline{S}\, :\, x\mapsto x^{-1}$	$\displaystyle \mathrm{reciprocal}$	(2)
$\displaystyle \overline{Z}\, :\, x\mapsto x+1$	$\displaystyle \mathrm{adding}\, 1$

$\displaystyle \mathbb {N}_{q}$	$\textstyle =$	$\displaystyle \left\{ \infty \right\} \cup \mathbb {Z}\left/ \left( q-1\right) \right.$	(3)
$\displaystyle F\doteq L_{\xi }\circ \psi \circ L_{\xi }^{-1}$		$\displaystyle F\left( s\right) =p\, s$
$\displaystyle S\doteq L_{\xi }\circ \overline{S}\circ L_{\xi }^{-1}$		$\displaystyle S\left( s\right) =-s$
$\displaystyle Z\doteq L_{\xi }\circ \overline{Z}\circ L_{\xi }^{-1}$		$\displaystyle \xi ^{Z\left( s\right) }=\xi ^{s}+1$

**Figure:** Carrying $\mathrm{GF}\! \left( q \right)$ structure over $\mathbb {N}_{q}\protect$ .
$\resizebox* {!}{7.5cm}{\includegraphics{velux.eps}}$

4.2 Elementary properties or Zech logarithms

$\displaystyle Z_{\xi }\circ F=F\circ Z_{\xi }$	$\textstyle i.\, e.$	$\displaystyle Z\left( p\, s\right) =p\, Z\left( s\right)$	(4)
$\displaystyle Z_{\xi }\circ S=Z_{\xi }+S$	$\textstyle i.\, e.$	$\displaystyle Z\left( S\left( s\right) \right) =Z\left( s\right) -s$

The $Z\, F$ formula is $\xi ^{Z\left( p\, s\right) }=\xi ^{p\, s}+1=\left( \xi ^{s}+1\right) ^{p}$ , i.e. a restatement of the Frobenius theorem, while the $Z\, S$ formula follows from : $\xi ^{Z\left( S\left( s\right) \right) +s}=\left( \xi ^{S\left( s\right) }+1\right) \xi ^{s}=1+\xi ^{s}$ .

From $\forall x\in \mathrm{GF}\! \left( q \right) \, :\, x=x+p$ , we get $Z^{-1}=Z^{p-1}$ and therefore :

$\displaystyle \mathrm{if}\quad p=2$	$\displaystyle Z^{-1}\left( \infty \right) =0\; ;\; Z^{-1}\left( s\right) =Z\left( s\right)$
$\displaystyle \mathrm{otherwise}$	$\displaystyle Z^{-1}\left( \infty \right) =\left( q-1\right) /2\; ;\; Z^{-1}\left( s\right) =r+Z\left( Z\left( s\right) -r\right)$	(5)
	$\displaystyle \mathrm{where}\quad \xi ^{r}=p-2$

Properties for $p=2$ are trivial. For odd primes, Fermat's theorem leads to $\left( \xi ^{\left( q-1\right) /2}\right) ^{2}=1$ . Since $\xi$ is primitive, $\mathrm{ord}\xi \neq \left( q-1\right) /2$ and $\xi ^{\left( q-1\right) /2}=-1$ . The $r$ in the last assertion is some multiple of $\left( q-1\right) /\left( p-1\right)$ , and can easily be calculated. Then $\xi ^{r+Z\left( Z\left( s\right) -r\right) }=\xi ^{r}\left( \xi ^{Z\left( s\r... ...i ^{Z\left( s\right) }+\xi ^{r}=\xi ^{s}+1+p-2=\xi ^{Z^{p-1}\left( s\right) }$ .

4.3 Cosets and coset mappings

We define the cyclotomic cosets in $\mathbb {N}_{q}$ as the orbits of $F\, :\, s\mapsto p\, s$ . Therefore, a given coset $C$ is the image under $L_{\xi }$ of a Frobenius-orbit in $\mathrm{GF}\! \left( q \right)$ . That mapping is $P_{\xi }$ -dependent, but the cosets themselves are canonical, i.e. independent from the choice of $\xi$ . As immediate results, it can be stated that $\char93 C\, \vert\, m$ and that $S$ maps a coset onto a coset of same size.

It follows from (4) that $Z$ maps also a coset onto a coset of same size. Adding together the equations $\xi ^{Z\left( s\right) }=\xi ^{s}+1$ over a whole coset leads to $\mathrm{Tr}\left( \xi ^{Z\left( s\right) }\right) \equiv \mathrm{Tr}\left( \xi ^{s}\right) +\char93 C\,{\rm mod}\,p$ . Therefore, $Z\left( C\right) =C$ can happen only when $p\, \vert\, \char93 C$ .

Let us now examine what happens to a given coset under the action of $Z$ and $S$ , i.e. under the action of the group $\left\langle Z,\, S \right\rangle$ they generate.

4.4 Frobenius cycles

As previously stated, $Z^{p}$ is nothing but the identity. From the primality of $p$ , it follows that the cardinal of the $\left\langle Z \right\rangle$ -orbit of a given coset is $1$ or $p$ .

4.5 Fibonacci cycles

Consider now the alternating use of $Z$ and $S$ , and the corresponding action $\overline{Z\, S}$ in $\mathrm{GF}\! \left( q \right)$ . We have $\overline{ZS}\left( x\right) =1+\frac{1}{x}$ . Therefore the iterations of $\overline{Z\, S}$ are leading to the same homographies as the expansion of $\left( 1+\sqrt{5}\right) /2$ , the golden mean, into continued fraction. As it is well-known, the coefficients of these $\mathbb {C}$ -homographies are the Fibonacci numbers $F_{0}=0$ , $F_{1}=1$ , $F_{n+2}=F_{n+1}+F_{n}$ . A straightforward expansion, leads to

$\begin{displaymath} \overline{Z\, S}\, x=1+\frac{1}{x}=\frac{x+1}{x}=\frac{F_{2}... ..._{n+1}\, x+F_{n}}=\frac{F_{n+2}\, x+F_{n+1}}{F_{n+1}\, x+F_{n}}\end{displaymath}$

Thus, for $p=2$ , $\overline{Z\, S}^{\, 3}\, x=\frac{3x+2}{2x+1}\equiv x$ and $\overline{Z\, S}^{\, 3}$ reduces to identity. Consequently, in a field $\mathrm{GF}\! \left( 2^{m} \right)$ , a $Z\, S$ cycle embodies $1,\, 2,\, 3$ , or, at most, $6$ cosets. Actually, quite all cycles of this kind embody $6$ cosets.

That property can be generalized to odd primes. let us denote by $u_{p}$ as in [5], the index of the smallest Fibonacci $F_{n}$ number that vanishes in $\mathbb {Z}\! \left/ p\right.$ . Then the length of a $Z\, S$ cycle is at most $2u_{p}$ since $\overline{Z\, S}^{\, u}\, x=\frac{\left( F_{u}+F_{u-1}\right) x+F_{u}}{F_{u}\, x+F_{u-1}}\equiv x$ . An upper bound for $u_{p}$ follows from the fact that a prime $p$ divides $F_{p-\left( 5/p\right) }$ , where $\left( 5/p\right)$ is the Legendre's symbol [Hardy, 4]. For small values of $p$ :

$p$	$u_{p}$	$F_{u}$	max. length
$2$	$3$	$2$	$6$
$3$	$4$	$3$	$8$
$5$	$5$	$5$	$10$
$7$	$8$	$21$	$16$

4.6 Flip-flop cycles

For odd primes, a third kind of coset-cycles appears when alternating $Z$ and $S$ with their reciprocals, i.e. when using $S$ , $Z$ , $S^{-1}$ , $Z^{-1}$ and so on. We get :

$\begin{displaymath} \overline{Z\, S}\, x=1+\frac{1}{x}=\frac{x+1}{x}\quad ;\quad... ...1}{x}\quad ;\quad \overline{Z^{-1}S\, Z\, S}\, x=\frac{-1}{x+1}\end{displaymath}$

With the conclusion that the cardinal of such a cycle divides 12, since the last homography is of order three in the complex plane.

4.7 The dimension of the $\left\langle Z,\, S \right\rangle$ group

Let us define a coset-orbit $CO$ as the orbit a given coset $C$ under the $\left\langle Z,\, S \right\rangle$ group, i.e. the set of all cosets that can be reached from $C$ when using any combination of $Z$ and $S$ . The cardinal of such a $CO$ is bounded by the cardinal of $\left\langle Z,\, S \right\rangle$ itself.

Its isomorphic image $\left\langle \overline{Z},\, \overline{S} \right\rangle$ is obviously a subgroup of the set of all homographic mappings $h\, :\, z\mapsto \left( a\, z+b\right) /\left( c\, z+d\right)$ where $a,\, b,\, c,\, d\in \mathbb {Z}\! \left/ p\right.$ and $a\, d-b\, c\neq 0$ , i.e. a subgroup of $\mathrm{PGL}_{2}\left( p \right)$ . For $p\neq 2$ , the projective group splits in two subclasses, according to the quadratic character of $a\, d-b\, c$ . Since $\det \left( z\mapsto z+1\right) =+1$ and $\det \left( z\mapsto 1/z\right) =-1$ , we have $\left\langle Z,\, S \right\rangle =\mathrm{PSL}_{2}\left( p \right)$ when $p\equiv 1\,{\rm mod}\,4$ and $\left\langle Z,\, S \right\rangle =\mathrm{PGL}_{2}\left( p \right)$ otherwise.

Therefore the cardinal of a $CO$ is at most $p^{3}-p$ , that bound being reduced by half in the $p\equiv 1\,{\rm mod}\,4$ case.

4.8 The hidden polyhedrons

When $p=3$ or $p=5$ , the $\left\langle Z,\, S \right\rangle$ group has a nice geometrical interpretation, by means of polyhedrons. Since the group $\mathrm{PGL}_{2}\left( 3 \right)$ is isomorphic to the octahedral group, the case $p=3$ can be illustrated by the Klein mapping [8], i.e. by placing the elements of $\left\langle Z,\, S \right\rangle$ at the vertices of that polyhedron obtained from an hexahedron (or an octahedron) by cutting off a (small) isosceles pyramid at each vertex.

The mapping [5], using the sides of that polyhedron, is incorrect. The side number is $36$ instead of $24$ . The $Z$ -cycles are mapped onto equilateral triangles, but not "near each vertex". And the octagons corresponding to the $\overline{Z\, S}$ cycles are skew, as shown Figure 3, and not "isosceles" since they cannot be the orbit of an 8-order element.

**Figure 3:** The $p=3$ case : a *skew* octagone generated by a $ZS$ cycle.
$\resizebox* {!}{0.2\textheight}{\includegraphics{octagone.eps}}$

When $p=5$ , it has been seen that $\left\langle S,\, Z \right\rangle \approx \mathrm{PSL}_{2}\left( 5 \right)$ . Therefore, that group can be mapped on the vertices of a truncated regular icosahedron (or, equivalently, at each "third" of the sides of a regular dodecahedron) as illustrated in Figure 4 (the labels ranging from $0$ to $15z$ are introduced in ***).

**Figure 4:** The $p=5$ case : a regular dodecahedron for the 60 elements of $\left\langle S,\, Z \right\rangle \protect$ .
$\resizebox* {!}{0.3\textheight}{\includegraphics{dodec4.eps}}$

4. Huber's paper revisited : the $\mathrm{PSL}_{2}\left( p \right) \protect$ group

4.1 Some mappings