# Elementary isomorphism between $\operatorname{PSL}(2,5)$ and $A_5$

At this Wikipedia page it is claimed that to construct an isomorphism between $\operatorname{PSL}(2,5)$ and $A_5$, "one needs to consider" $\operatorname{PSL}(2,5)$ as a Galois group of a Galois cover of modular curves and consider the action on the twelve ramified points. While this is a beautiful construction, I wonder if this really is necessary. Is there a construction of a map that takes a representative matrix of a class in $\operatorname{PSL}(2,5)$ and uses some relatively simple computation to produce a permutation in $S_5$ that can be shown to be even? I don't mind if describing the construction and providing the verification that it is well-defined and does what it should takes several pages. I'd just like to think that it's possible.

• It may depend if you want a geometric construction or an algebraic one. Since a Sylow $2$-subgroup of ${\rm PSL}(2,5)$ is a Klein $4$-group, Burnside's normal $p$-complement theorem guarantees that a Sylow $2$-normalizer has order $12,$ so that ${\rm PSL}(2,5)$ has $5$ Sylow $2$-subgroups, yielding a homomorphism from ${\rm PSL}(2,5)$ to $S_5$ with trivial kernel and image of order $60.$ Commented Dec 23, 2011 at 20:42
• (somewhat) related: math.stackexchange.com/questions/57986/… Commented Dec 23, 2011 at 20:48
• Since those Sylow 2-subgroups have order 4, presumably there might be a relatively simple description of representative matrices that generate these groups? That would be explicit enough to satisfy me. Commented Dec 23, 2011 at 20:50
• I would then also be happy with an extremely explicit construction of this covering. Commented Dec 23, 2011 at 20:56

One way to see this, which can be found in Galois's letter to Chevalier that he wrote on the night before his death, is that $$G = PSL(2,5)$$ contains a maximal subgroup $$H$$ of index $$5$$. The action of $$G$$ on $$G/H$$ is faithful (because $$G$$ is simple), and so we get an embedding of $$G$$ into $$S_5$$. Since $$S_5$$ doesn't contain many subgroups of order 60, we are done.

Galois more generally considers the action of $$PSL(2,p)$$ on the fibres of the modular curve $$X_0(p)$$ over $$X_0(1)$$ (which have $$p+1$$ points generically), and from this point of view sees that $$PSL(2,p)$$ can appear as the Galois group of a degree $$p+1$$ equation (the equation cutting out the fibre over a typical $$j$$-invariant in $$X_0(1)$$). He asks whether we can replace this degree $$p+1$$ equation by a degree $$p$$ one, and observes that this is possible for $$p = 5,7,11$$ (i.e. these are the primes for which $$PSL(2,p)$$ has an index $$p$$ subgroup).

It's quite amazing to see just how much Galois understood!

Geometric description: Both are isomorphic to the group of symmetries of the icosahedron/dodecahedron. To show this, I will five color the faces of the icosahedron, and label the faces of the dodecahedron with $\{ 0,1,2,3,4,\infty \}$, so the symmetry group acts on the $5$ colors by $A_5$ and acts on the $6$ elements of $\mathbb{P}^1(\mathbb{F}_5)$ by $PSL(2,5)$. Cutout out the two images below and tape them together to see the construction:

• Alternatively, there is a way (two, actually) to partition the vertices of a dodecahedron in groups of four in such a way that each part consists of the vertices of a regular tetrahedron. It is not difficult to see that a motion of the solid cannot give a transposition of two of the tetrahedra. Commented Mar 12, 2012 at 19:16
• Yup. The colors here are vertices of the tetrahedra Mariano speaks of. I've always thought images of the tetrahedra were too cluttered to understand and that this might be clearer. It looked better in my head than on paper, though. Commented Mar 12, 2012 at 19:20
• Google will find modular origami instructions to build the five tetrahedra. It takes a little patience to fold the pieces and quite a bit of ingenuity to assemble, but the result is quite extraordinary. Drop by at my office if you are ever in the neighborhood and I'll let you play with mine :) Commented Mar 12, 2012 at 19:24
• Feel free to edit in a photo of your desktoys! Commented Mar 12, 2012 at 19:27

I know two good descriptions, which I'll put in two answers. Here is the algebraic one:

The group $PGL(2,5)$ acts on $\mathbb{P}^1(\mathbb{F}_5)$, a set which has $6$ elements. There are $15$ permutations of $\mathbb{P}^1(\mathbb{F}_5)$ which have order $2$ and no fixed points -- for example, $0 \leftrightarrow \infty$, $1 \leftrightarrow 2$, $3 \leftrightarrow 4$. Of those permutations, $10$ are induced by elements of $PGL(2,5)$. For example, the above map is $z \mapsto 2/z$. Conjugation by $PGL(2,5)$ permutes the other $5$ involutions, and this permutation gives an isomorphism $PGL(2,5) \cong S_5$.

• Note that in the terminology of John Baez (see math.ucr.edu/home/baez/six.html), the icosahedron contains 5 true crosses and 10 skew crosses; each of those 15 crosses naturally corresponds to an order 2 permutation of the axes with no fixed points, and the 5 such involutions not induced by PGL(2,5) correspond nicely to the 5 true crosses, and so this conjugation action is the same as the icosahedron's symmetries acting on those 5. Commented Mar 4, 2021 at 0:34

Yet another way: I will show that both of them are isomorphic to the group of rotational symmetries of the dodecahedron. Let $H$ be the rotational symmetry group of the dodecahedron.

$H$ embeds in $SO(3)$, using entries in $\mathbb{Z}[\tau]$ where $\tau$ is the golden ratio $\frac{1 + \sqrt{5}}{2}$. Since these are orthogonal matrices, they preserve the quadratic form $x^2+y^2+z^2$.

First, consider the ideal $\langle \sqrt{5} \rangle = \langle 2 \tau-1 \rangle$ in $\mathbb{Z}[\tau]$. Reducing our matrices modulo this ideal, we get matrices in $SL_3(\mathbb{F}_5)$ preserving the quadratic form $x^2+y^2+z^2$. The conic $x^2+y^2+z^2=0$ (in homogenous coordinates) has points over $\mathbb{F}_5$, namely, the $6$ permutations of $(0:1:2)$. So the matrices act on this conic by matrices in $PGL_2(\mathbb{F}_5)$. This gives a map $H \to PGL_2(\mathbb{F}_5)$, and it is not bad to show it is an isomorphism $H \to PSL_2(\mathbb{F}_5)$.

Now, consider the ideal $\langle 2 \rangle$ in $\mathbb{Z}[\tau]$. The quotient is the field with $4$-elements, so we get a representation of $H$ by matrices in $SL_3(\mathbb{F}_4)$ preserving $x^2+y^2+z^2 = (x+y+z)^2$. Thus, these matrices preserve the plane $x+y+z=0$. We get an action on the set of $5$ lines in this plane. This gives a map $H \to S_5$, and it isn't hard to see that it is an ismorphism $H \to A_5$. Remark: We have also just shown $H \cong PGL_2(\mathbb{F}_4)$ (which is the same as $PSL_2( \mathbb{F}_4)$).

Nothing interesting seems to happen if we reduce modulo $\langle 3 \rangle$, I'm sorry to say. We get an action of $H$ on $\mathbb{P}^1(\mathbb{F}_9)$, but I can't find much to say about it.

• I learnt this construction from someone else: Let $2A_5$ denote the binary icosahedral group; there's a faithful rep $2A_5 \to SU(2)$ which has entries in $\Bbb Z[1/2, i, \phi]$ where $\phi$ is the golden ratio, which if we reduce mod $3$ is $\Bbb F_3[i]$. That buys us a rep $2A_5 \to \text{SL}_2(\Bbb F_3[i])$, and quotienting out the center gives $A_5 \to \text{PSL}_2(\Bbb F_3[i]) \cong \text{PSL}_2(\Bbb F_9)$, which seems to be your action perhaps? This is also faithful by simplicity of the domain. Commented Nov 3, 2020 at 20:34
• What this says is that there is a copy of $A_5$ sitting inside $\text{PSL}_2(\Bbb F_9)$ which is index $6$, so the permutation representation gives a map $\text{PSL}_2(\Bbb F_9) \to S_6$. By simplicity of the domain again we get an isomorphism to $A_6$, explaining a different exceptional isomorphism $A_6 \cong \text{PSL}_2(\Bbb F_9)$ Commented Nov 3, 2020 at 20:35

The order of $G=PSL(2,5)$ is 60. A Sylow 2-subgroup $P$ is generated by the cosets represented by $$A=\pmatrix{2&0\cr0&3\cr}\quad\text{and}\quad B=\pmatrix{0&2\cr3&0\cr}.$$ It is easy to check that $P$ is normalized by
$$C=\pmatrix{1&2\cr1&3\cr}$$ (conjugation by $C$ cycles the involutions in a 3-cycle $B\mapsto A\mapsto AB=BA \mapsto B$). The order of $C$ is three. As $P$ is not normal in $G$, we can deduce that $N(P)$ has order 12. Thus there are 5 Sylow 2-subgroups. Therefore we get a homomorphism from $G$ to $S_5$ from the conjugation action of $G$ on the 2-Sylow subgroups. If you believe that $G$ is simple, then rest follows as in Geoff Robinson's comment. Even without using that fact at this point it would suffice to prove that this homomorphism is injective. Or equivalently that the intersection of the conjugates of $N(P)$ is trivial.

The other Sylow 2-subgroups can be gotten by conjugating $P$ with powers of $$D=\pmatrix{1&1\cr0&1\cr}.$$

• I wasn't really assuming ${\rm PSL}(2,5)$ simple in my comment, it's clear that the homomorphism into $S_5$ is injective by a direct argument of the kind you outlined. Commented Dec 24, 2011 at 1:45