Endomorphism structure of Klein four-group

This article gives specific information, namely, endomorphism structure, about a particular group, namely: Klein four-group.
View endomorphism structure of particular groups | View other specific information about Klein four-group

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] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
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] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
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] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
whole group	2	trivial group	0	Yes	Yes	1	$\begin{pmatrix}0 & 0 \\ 0 & 0 \\\end{pmatrix}$
Total	--	--	--	--	--	16