# Subgroup structure of elementary abelian group:E8

## Contents

View subgroup structure of particular groups | View other specific information about elementary abelian group:E8

We consider here the elementary abelian group:E8. For notational simplicity, we consider this group as $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$ and represente its elements as ordered 3-tuples with entries from the integers mod 2 with coordinate-wise addition mod 2.

## Tables for quick information

### Table classifying subgroups up to automorphisms

Automorphism class of subgroups Representative subgroup Isomorphism class Order of subgroups Index of subgroups Dimension as vector space over field:F2 (= log of order to base 2) Codimension as vector space over field:F2 (= log of index to base 2) Number of conjugacy classes Size of each conjugacy class Total number of subgroups Quotient group
trivial $\{ (0,0,0) \}$ trivial group 1 8 0 3 1 1 1 elementary abelian group:E8
Z2 in E8 $\{ (0,0,0), (1,0,0) \}$ cyclic group:Z2 2 4 1 2 7 1 7 Klein four-group
V4 in E8 $\{ (0,0,0), (1,0,0), (0,1,0), (1,1,0) \}$ Klein four-group 4 2 2 1 7 1 7 cyclic group:Z2
whole group all elements elementary abelian group:E8 8 1 3 0 1 1 1 trivial group
Total (4 rows) -- -- -- -- -- -- 16 -- 16 --

## Lattice of subgroups

The lattice of subgroups is bounded at both ends by the trivial subgroup and whole group. Ignoring these, the rest of the lattice can be viewed as a bipartite graph between the subgroups of order two and the subgroups of order four. If we think of elementary abelian group:E8 as a vector space over field:F2, then the subgroups of order two are 1-dimensional affine subspaces (lines) and the subgroups of order four are 2-dimensional affine subspaces (planes). The containment relation of these is captured by looking at the projective plane over field:F2, which is a geometry where:

• the points are the one-dimensional affine subspaces or lines (i.e., the order two subgroups)
• the lines are the two-dimensional affine subspaces or planes (i.e., the order four subgroups), and
• the incidence relation is defined by containment of the point (actually, a line) inside a line (actually, a plane)

The projective plane over the field of two elements is a Fano plane and its picture is below. The thickened dots represent the points (one-dimensional affine subspaces) and the lines (including one drawn as a circle) represent the lines. A point is incident to a line if and only if the corresponding order two subgroup is incident to the corresponding order four subgroup.