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|