Endomorphism structure of dihedral group:D8

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This article gives specific information, namely, endomorphism structure, about a particular group, namely: dihedral group:D8.
View endomorphism structure of particular groups | View other specific information about dihedral group:D8

This article is about the structure of endomorphisms of dihedral group:D8, which we will take as having the following presentation:

\langle x,a \mid a^4 = x^2 = e, xax^{-1} = a^{-1} \rangle

where e denotes the identity element.

Summary of information

Construct Value Order Second part of GAP ID (if group)
endomorphism monoid  ?  ?
automorphism group dihedral group:D8 8 3
inner automorphism group Klein four-group 4 2
extended automorphism group direct product of D8 and Z2 16 11
quasiautomorphism group direct product of D8 and Z2 16 11
1-automorphism group direct product of S4 and Z2 48 48
outer automorphism group cyclic group:Z2 2 1

Description of automorphism group

Inner automorphisms

For the actual description of automorphisms, we use the left action convention, so conjugation by g is the map h \mapsto ghg^{-1}. With the right action convention, what we call conjugation by g becomes conjugation by g^{-1}.

For every inner automorphism, there are two elements that give rise to it via the action by conjugation. Both these elements together form a coset of the center. The columns have been arranged so that all elements of a conjugacy class are in adjacent columns to each other. From this, you can notice that the inner automorphisms permute elements within their conjugacy class.

Automorphism can be thought of as conjugation by (first element) Automorphism can be thought of as conjugation by (second element) Image of e Image of a^2 Image of a Image of a^3 Image of x Image of a^2x Image of ax Image of a^3x
e a^2 e a^2 a a^3 x a^2x ax a^3x
a a^3 e a^2 a a^3 a^2x x a^3x ax
x a^2x e a^2 a^3 a x a^2x a^3x ax
ax a^3x e a^2 a^3 a a^2x x ax a^3x

The inner automorphism group itself is isomorphic to a Klein four-group. The multiplication table, viewed in terms of the corresponding cosets of the center, is below:

Element/element \{ e, a^2 \} \{ a, a^3 \} \{ x, a^2x \} \{ ax, a^3x \}
\{ e, a^2 \} \{ e, a^2 \} \{ a, a^3 \} \{ x, a^2x \} \{ ax, a^3x \}
\{ a, a^3 \} \{ a, a^3 \} \{ e, a^2 \} \{ ax, a^3x \} \{ x, a^2x \}
\{ x, a^2x \} \{ x,a^2x \} \{ ax, a^3x \} \{ e, a^2 \} \{ a, a^3 \}
\{ ax, a^3x \} \{ax, a^3x \} \{ x, a^2x \} \{ a, a^3 \} \{ e, a^2 \}