# Endomorphism structure of Klein four-group

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## Contents

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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) $M(2,2)$: matrix ring of $2 \times 2$ matrices over field:F2 16 14 (for additive group of endomorphism ring)
automorphism group symmetric group:S3 (isomorphic to $GL(2,2)$) 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 $\mathbb{F}_2$-linear maps from this vector space to itself. These endomorphisms are described as $2 \times 2$ matrices over $\mathbb{F}_2$, 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 $\mathbb{F}_2$.

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 $\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}$
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 $\begin{pmatrix}0 & 0 \\ 0 & 0 \\\end{pmatrix}$
Total -- -- -- -- -- 16