Endomorphism structure of Klein four-group
This article gives specific information, namely, endomorphism structure, about a particular group, namely: Klein four-group.
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This article is about the structure of endomorphisms (and in particular automorphisms) of the Klein four-group, which is the elementary abelian group of order 4, or equivalently, the direct product of two copies of cyclic group:Z2.
Summary of information
Construct | Value | Order | Second part of GAP ID (if group) |
---|---|---|---|
endomorphism ring (applicable since it's an abelian group) | : matrix ring of matrices over field:F2 | 16 | 14 (for additive group of endomorphism ring) |
automorphism group | symmetric group:S3 (isomorphic to ) | 6 | 1 |
inner automorphism group | trivial group | 1 | 1 |
extended automorphism group | symmetric group:S3 | 6 | 1 |
quasiautomorphism group | symmetric group:S3 | 6 | 1 |
1-automorphism group | symmetric group:S3 | 6 | 1 |
outer automorphism group | symmetric group:S3 | 6 | 1 |
Description of endomorphism monoid
The Klein four-group can be viewed as a two-dimensional vector space over field:F2. Moreover, endomorphisms of this as a group are precisely the same as -linear maps from this vector space to itself. These endomorphisms are described as matrices over , with endomorphism composition given by matrix multiplication. Note that this identification depends on a choice of basis for the group as a vector space over .
Below is the complete list of endomorphisms, grouped together by similarity type of matrices (which means by conjugacy via automorphisms):
Kernel of endomorphism as group | Rank of kernel as vector space | Image of endomorphism as group | Rank of image as vector space | Nilpotent? | Retraction? | Number of endomorphisms | List of matrices for endomorphisms |
---|---|---|---|---|---|---|---|
trivial group | 0 | whole group | 2 | No | Yes | 1 | |
trivial group | 0 | whole group | 2 | No | No | 5 | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
cyclic group:Z2 (Z2 in V4) | 1 | cyclic group:Z2 (Z2 in V4) | 1 | Yes | No | 3 | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
cyclic group:Z2 (Z2 in V4) | 1 | cyclic group:Z2 (Z2 in V4) | 1 | No | Yes | 6 | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
whole group | 2 | trivial group | 0 | Yes | Yes | 1 | |
Total | -- | -- | -- | -- | -- | 16 |