Endomorphism structure of dihedral group:D8
This article gives specific information, namely, endomorphism structure, about a particular group, namely: dihedral group:D8.
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This article is about the structure of endomorphisms of dihedral group:D8, which we will take as having the following presentation:
where denotes the identity element.
Summary of information
|Construct||Value||Order||Second part of GAP ID (if group)|
|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
For the actual description of automorphisms, we use the left action convention, so conjugation by is the map . With the right action convention, what we call conjugation by becomes conjugation by .
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||Image of||Image of||Image of||Image of||Image of||Image of||Image of|
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: