Quatre groupes à double vie

Let $G$ be a finite group. A life of $G$ is an isomorphism $\mu\colon G\to\Gamma$ onto a member $\Gamma$ of one of the two families; through $\mu$ the natural sets of $\Gamma$ are objects of $G$. A double life of $G$ is a pair of lives $\mu_1\colon G\to\Gamma_1$, $\mu_2\colon G\to\Gamma_2$ in different members of the families. It is built when the isomorphism $\mu_2\mu_1^{-1}\colon\Gamma_1\to\Gamma_2$ is given explicitly. Its dictionary lists, for each object of $G$, the natural sets of each life that are incarnations of it.

Among the simple groups $\PSL(n,q)$, $(n,q)\neq(2,2),(2,3)$, and $A_m$, $m\ge5$, two groups have the same order only at 60: $\PSL(2,4)$, $\PSL(2,5)$, $A_5$; at 168: $\PSL(2,7)$, $\PSL(3,2)$; at 360: $\PSL(2,9)$, $A_6$; and at 20160: $\PSL(3,4)$, $\PSL(4,2)$, $A_8$. The groups of each order are isomorphic, with the single exception that $\PSL(3,4)$ is not isomorphic to $\PSL(4,2)\cong A_8$.

Matching stabilizers through $\theta$, the natural sets of the two lives are incarnations of four objects. For $A_5$ the natural sets are the subsets of the letters, the pairs of disjoint pairs and the pentagons; for $\PSL(2,5)$ the subsets, the bisections $3|3$ and the partitions of $\Proj^1(\F_5)$ into three pairs.

Size 5, stabilizer $A_4$: the letters; the orbit of 5 among the 15 partitions into three pairs. Size 6, $D_{10}$: each of the two orbits of 6 pentagons; the points. Size 10, $S_3$: the pairs of letters; the bisections $3|3$, each of the two orbits of 10 triples, and the orbit of 10 partitions into three pairs. Size 15, $V_4$: the pairs of disjoint pairs; the pairs of points.

The objects of sizes 1, 12, 20, 30 and 60, with stabilizers $A_5$, $C_5$, $C_3$, $C_2$ and 1, are among these natural sets in neither life.

$\GL(3,2)$ has eight Sylow 7-subgroups and acts on them by conjugation. Exactly 336 bijections from this set onto $\Proj^1(\F_7)$ carry the resulting permutation group onto $\PSL(2,7)$ acting by Möbius transformations, and each defines an isomorphism $\GL(3,2)\to\PSL(2,7)$.

$\PSL(2,9)$, acting on the ten points of $\Proj^1(\F_9)$, has twelve subgroups isomorphic to $A_5$, in two conjugacy classes of six. For each class, the action on the six cosets is an isomorphism onto $A_6$. The two isomorphisms differ by an automorphism of $A_6$ that carries 3-cycles to products of two disjoint 3-cycles, so it is induced by no permutation of the six letters. Through either isomorphism the stabilizer of each point of $\Proj^1(\F_9)$, of order 36, fixes exactly one bisection $3|3$ of the six letters, and this gives an equivariant bijection from the ten points onto the ten bisections.

(1) $\GL(4,2)$ acts faithfully on $(\Lambda^2V,Q)$ and $S_8$ on $(E,q)$, and the two quadratic spaces are isometric, by an isometry $\varphi$ that matches hyperbolic bases.

(2) $\varphi$ carries the image of $\GL(4,2)$ onto the image of $A_8$, so $g\mapsto$ the even permutation inducing $\varphi\,\Lambda^2(g)\,\varphi^{-1}$ is an isomorphism $\PSL(4,2)=\GL(4,2)\to A_8$.

(3) Under it the 35 lines of $\mathrm{PG}(3,2)$ are the 35 bisections $4|4$ of the letters; the 15 points and the 15 planes are the two $A_8$-orbits on the 30 structures of affine 3-space on the letters, the Steiner systems $S(3,4,8)$; the 28 nonsingular vectors are the 28 pairs of letters. The stabilizer of a letter, a group $A_7$, acts 2-transitively on the 15 points.

(4) An odd permutation of the letters carries the structures of the points onto those of the planes: the outer automorphism of $A_8$ induced by $S_8$ is the duality of $\mathrm{PG}(3,2)$.

Let $L$ have good reduction at $\mathfrak P$ and let $(\sigma,\alpha)$ be a twisted Galois symmetry. Then $L$ has good reduction at $\sigma\mathfrak P$, and (a) $(V_{\sigma\mathfrak P})^{\alpha}\cong\bar\sigma V_{\mathfrak P}$. In particular:

(b) split type: if $\sigma\notin D(\mathfrak P)$, there is a $\bar\sigma$-semilinear bijection $f\colon V_{\mathfrak P}\to V_{\sigma\mathfrak P}$ with $f\rho_{\mathfrak P}(g)=\rho_{\sigma\mathfrak P}(\alpha(g))f$, a seam over $\alpha$ to the residue at another prime over $p$;

(c) inert type: if $\sigma\in D(\mathfrak P)\setminus I(\mathfrak P)$, there is a $\bar\sigma$-semilinear bijection $\Phi$ of $V_{\mathfrak P}$ with $\Phi\rho_{\mathfrak P}(g)\Phi^{-1}=\rho_{\mathfrak P}(\alpha(g))$, which can be chosen linear if and only if $\bar\sigma$ fixes every trace;

(d) ramified type: if $\sigma\in I(\mathfrak P)$, there is a linear $M$ with $M\rho_{\mathfrak P}(g)M^{-1}=\rho_{\mathfrak P}(\alpha(g))$, normalizing $\rho_{\mathfrak P}(G)$; if $G$ is perfect, $\ker\rho_{\mathfrak P}$ central and $\alpha$ outer, $M$ is not a scalar multiple of an element of $\rho_{\mathfrak P}(G)$;

(e) duality: if moreover $\sigma\circ\chi=\bar\chi$, then at every prime of good reduction $\alpha$ carries the residue to its dual.

Each lattice is twisted Galois stable, with $\Gamma\to\operatorname{Out}(G)$ onto. Klein’s lattice, rank 3 over $\Q(\sqrt{-7})$: at $2=\mathfrak p\bar{\mathfrak p}$, $c$ splits, the two Fano planes are dual and $\alpha_c$ is a polarity, a seam from points to lines; at 7, $c$ ramifies and $\alpha_c$ is odd on the eight points of $\Proj^1(\F_7)$, in $\PGL(2,7)\setminus\PSL(2,7)$; at 3, $c$ is inert, $G\subset\mathrm{PSU}(3,3)$, and $\alpha_c$ is the Frobenius of $\F_9$.

The icosians: at 2 and at 3 the Galois element is inert and of field type; at $\sqrt5$ it ramifies and is a linear map of non-square determinant. $E_8$ over $\Z[\omega]$: at $\sqrt{-3}$, $c$ ramifies, the form is alternating and $\alpha_c$ a similitude of multiplier $-1$, on the 40 points and 40 lines of $W(3)$; at 2 it is inert, hermitian and Frobenius-semilinear. Valentiner’s lattice: $\Gamma\cong\operatorname{Out}(A_6)$; at 2 the six letters are a hyperoval of $\mathrm{PG}(2,4)$ and $\alpha_c$ is a duality carrying its six passant lines to its six points; at 3, on the conic $\Proj^1(\F_9)$, $c$ ramifies and is linear. $\PSL(2,8)$ and $\mathrm U_3(3)$: at a life’s prime there is no good reduction, and the law holds on the composition factors.

For $\PSL(2,7)$ and $A_6$, the outer automorphism that exchanges two dual objects of one life and is a non-square Möbius map in the other is $\alpha_c$, $c$ the complex conjugation of the CM field $K=\Q(\chi)$ of Klein’s, respectively Valentiner’s, lattice. The prime of the dual life splits in $K/K^+$ and the prime of the Möbius life ramifies in $K/K^+$. For $A_5$, $\Q(\chi)$ is totally real, and its outer automorphism is of field type at 2 and diagonal at 5. For $A_8$ the pattern is outside the law.