Subgroup structure of elementary abelian group:E8
This article gives specific information, namely, subgroup structure, about a particular group, namely: elementary abelian group:E8.
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We consider here the elementary abelian group:E8. For notational simplicity, we consider this group as 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 | trivial group | 1 | 8 | 0 | 3 | 1 | 1 | 1 | elementary abelian group:E8 | |
| Z2 in E8 | cyclic group:Z2 | 2 | 4 | 1 | 2 | 7 | 1 | 7 | Klein four-group | |
| V4 in E8 | 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.
