arXiv:1709.05532v2 [] 15 Feb 2018

The Unknown Subgroup of Aut(E8)
Majid Butler, Sandernisha Claiborne, Tomme Denney,
De’Janeke Johnson and Tianna Robinson

McDonogh 35 High School
The authors were supported by the Student Research
Fellowship program at McDonogh 35 High School in New Orleans.

We give several descriptions of the maximal subgroup 2A9 <
Aut(E8 ), the simplest of which is: 2A9 permutes nine scale copies
of E8 generated by the 9 · 240 = 2160 norm 4 vectors.
August 10, 2017

1

INTRODUCTION

In the 150 years since it was shown to exist, the E8 lattice has been one of
the most studied objects in mathematics. Despite this scrutiny a significant
gap in our knowledge of E8 remains; the subgroup 2A9 , maximal in Aut(E8 ),
has defied simple description. In fact the ATLAS characterizes every maximal
subgroup of the simple group O8 + (2), save one: only 2A9 exhibits blank entries
across the page. The purpose of our brief paper is to illuminate this one final
group.
Our results can be summarized in three ways, based on the following three
perspectives: looking at E8 modulo 2; looking at the lattice’s norm 2 vectors;
and looking at the norm 4 vectors. These perspectives reveal 2A9 as:
(i) the automorphisms of E8 that stabilize a partition of the 135 isotropic
points of E8 /2E8 into nine disjoint isotropic 4-spaces of 15 points each;1 or
(ii) the automorphisms of E8 that stabilize a 9 x 15 array of coordinate
frames such that
1 The subgroup that fixes one of the nine 4-spaces permutes the other eight 4-spaces while
it permutes the 15 points of the fixed space, so the isomorphism A8 ∼
= L4 (2) is visible within
E8 .

1

a) each of the 120 root vectors of E8 appears once in each row, as one of
eight vectors in one of 15 coordinate frames, while
b) each pair of orthogonal root vectors appear together in exactly one of the
135 frames; or
(iii) the automorphisms of E8 that stabilize a partition of the lattice’s 2160
norm 4 vectors into nine sets of 240, where each set generates a scale copy of
the E8 lattice itself.

2

MODULO 2

We begin by showing that the 135 isotropic points in an 8-space over F2 (which
admits a quadratic form of Witt defect 0) can be arranged into nine disjoint
isotropic 4-spaces that are permuted by a subgroup A9 < O8 + (2).
There are 270 isotropic 4-spaces in the 8-space, and O8 + (2).2 permutes them
transitively while the subgroup O8 + (2) of index 2 permutes them in two orbits
of 135 each, with stabilizer Stab(V1 ) ∼
= 26 A8 . This stabilizer acts on the other
2
134 4-spaces in orbits of size 64 (which have trivial intersection with V1 ) and 70
(which intersect V1 in a 2-space.) If V2 is a 4-space in the 64-orbit then Stab(V1 )
∩ Stab(V2 ) is a subgroup isomorphic to A8 that permutes the remaining 133
4-spaces in orbits of sizes 28 + 35 + 35 + 35 (which intersect {V1 , V2 } in spaces
of dimension {0, 0}, {0, 2}, {2, 0} and {2, 2}, respectively.)
The 28 4-spaces disjoint from both V1 and V2 can be labeled by the duads
of eight letters. The subgroups A7 < A8 permute the seven 4-spaces (Va∗ ) =
{Vab , Vac , . . ., Vah } that share a letter, and the subgroups A6 < A8 stabilize
two 7-sets A = (Va∗ ) and B = (Vb∗ ), so that each A6 also fixes a unique third
4-space Vab while permuting the six other 4-spaces in A and in B.
We use this simple fact to construct A9 . We begin with two disjoint 4-spaces
V1 and V2 (whose pointwise stabilizer is A8 ) and we choose a subgroup A6 < A8
that permutes six 4-spaces {Vac , . . ., Vah } and fixes a unique 4-space Vab ; note
that A6 extends to an A7 < A8 that permutes the seven 4-spaces {Vab , Vac , . . .,
Vah }. But since V1 , V2 and Vab are mutually disjoint, we can start instead with
the A8 that fixes V1 and Vab ; since this group contains the above A6 (which fixes
V1 , V2 and Vab and permutes the six 4-spaces {Vac , . . ., Vah }) we can extend
A6 to an A7 subgroup that still fixes V1 and Vab but now permutes the seven
4-spaces {V2 , Vac , . . ., Vah }.3 Or likewise we can start with the A8 that fixes V2
and Vab to obtain an A7 subgroup that permutes the seven 4-spaces {V1 , Vac ,
. . ., Vah }. These three A7 subgroups together generate a subgroup of O8 + (2)
that stabilizes the nine 4-spaces {V1 , V2 , Vab , Vac , . . ., Vah } and projects onto
A9 ; in fact this subgroup cannot be any larger than A9 since otherwise its onepoint stabilizer would be larger than A8 , which is already maximal in Stab(V1 )
2 Unless otherwise noted, throughout the remainder of this paper the term “4-space” will
mean one of the 135 isotropic 4-spaces within the orbit in question.
3 A acts on the 28 duads of an 8-letter set in orbits of sizes 1+6+6+15 and acts on the
6
35 bisections in orbits of sizes 15+20, so {Vac , . . ., Vah }, an orbit of size 6, must lie in the
28-orbit for A8 ∼
= Stab(Vi ) ∩ Stab(Vab ), i = {1,2}.

2

∼
= 26 A8 .
Finally, since V1 and V2 are disjoint and since A9 is multiply transitive on
the nine 4-spaces, these nine 4-spaces are pairwise disjoint and therefore exhaust
the 9·15 = 135 isotropic vectors.

3

NORM 2

This arrangement of the 135 isotropic vectors into nine isotropic 4-spaces partitions the 120 root vectors of E8 into nine sets of 15 coordinate frames each,
with properties (ii)(a) and (ii)(b) above.
Let V be one of the nine 4-spaces and consider a 3-space W ⊂ V. The space
W⊥ is 5-dimensional and contains W, so W⊥ /W is a 2-space with three non-zero
vectors. One of these vectors completes W into V (and so is isotropic) and at
least one of the other two vectors is non-isotropic, since W⊥ cannot be a totally
isotropic 5-space. If r is a non-isotropic vector4 then the eight non-isotropic
vectors {r + w, w ∈ W} lift to eight different root vectors {r1 , . . ., r8 } of E8 .
Since the sum ri + rj of any two of these root vectors projects to an (isotropic)
vector of W, this E8 vector has norm 4, so that ri and rj are orthogonal. In
other words, these eight root vectors form a coordinate frame.
Furthermore, no non-isotropic vector r can lie in W1 ⊥ and W2 ⊥ for distinct
3-spaces W1 , W2 of V since otherwise r ∈ W1 ⊥ ∩ W2 ⊥ = V. Thus the 15 3-spaces
of V yield 15 frames in which each of the 120 root vectors appears only once.
The nine different 4-spaces {V1 , . . ., V9 } therefore give us nine different
partitions of the 120 root vectors into 15 coordinate frames. No pair of root
vectors {ra , rb } may lie in two of these 135 coordinate frames (for 4-spaces Vi
and Vj , say) since then ra + rb would project to a vector in Vi ∩ Vj = {0}.
But since there are 120 · 63/2 = 3780 pairs {ra , rb } of orthogonal root vectors,
and since the 135 8-vector frames yield 135 · 82 = 3780 such pairs, each pair
appears in exactly one coordinate frame. Note also that since a pair {ra , rb } of
orthogonal root vectors gives rise to four norm 4 vectors {±ra ±rb }, each norm
4 vector is derived from two root vectors in 3780 · 4/2160 = 7 different ways.

4

NORM 4

Suppose we begin with the “norm 2” viewpoint: we have a 9 x 15 array of
coordinate frames in which the 15 frames in each row were obtained from the
15 3-spaces of a 4-space. We use this array to partition the 2160 norm 4 vectors
into nine scale copies of E8 , as follows.
Each coordinate frame of eight vectors {r1 , . . ., r8 } yields ±56 norm 4 vectors
{±ri ±rj }, and a row’s 15 frames therefore yield ±56·15/7 = ±120 norm 4
vectors. But since a row was obtained from an isotropic 4-space its norm 4
vectors have even inner product with each other, so we can divide the natural
4 The third vector is isotropic and extends W to an isotropic 4-space in the other orbit of
135, leading to another description of 2A9 based on the same 9 x 15 arrangement of 3-spaces.

3

inner product by 2 to obtain an integral rank 8 lattice generated by ±120 norm 2
vectors. If we fix one of the 15 coordinate frames in this row then the half-scale
inner product turns this frame’s eight root vectors {ri } into an orthonormal
basis, in which case the ±56 vectors {±ri ±rj } generate a D8 lattice. The
remaining ±64 vectors in the other 14 frames are norm 2 elements of D8 ∗ , any
one of which extends D8 to E8 . And since each row contains 240 of the 2160
norm 4 vectors, the nine rows partition the norm 4 vectors into nine scale copies
of E8 .
Finally, if the 2160 norm 4 vectors of E8 are partitioned into nine doublescale copies of E8 , then the 240 vectors in each copy have even inner products
and so form a totally isotropic subspace in E8 /2E8 . Hence these 240 vectors
project onto at most 15 points in E8 /2E8 —an isotropic space cannot be larger—
—and all nine copies of E8 project onto no more than 135 non-zero points. So
equality must hold: the nine scale copies of E8 generate a partition of the 135
isotropic vectors into nine disjoint 4-spaces of 15 points each, which is exactly
our original perspective.

Contacts: majidhbutler54@gmail.com, nisha.claiborne@gmail.com,
tomme.denney32@gmail.com, jdejaneke@yahoo.com, and
tiannarobinson71401@gmail.com
References
[1] J.H. Conway, R.T. Curtis, S.P Norton, R.A. Parker,
and R.A. Wilson. An Atlas of Finite Groups. Oxford
University Pressy, 198 Madison Avenue New York,
NY 10016, 1985.

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