Endomorphism structure of 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.

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 $$g^{-1}$$. However, the data in the table below is the same for both left and right action conventions because, since every square is in the center, every element and its inverse are in the same coset of the center.

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.

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:

Outer automorphisms and outer automorphism classes
The outer automorphism group is cyclic group:Z2, which means that there is only one non-identity coset of the inner automorphism group in the automorphism group. This non-identity outer automorphism class preserves the conjugacy class $$\{ a, a^3 \}$$, which is the unique conjugacy class of elements of order four. It interchanges the other two conjugacy classes. The precise action at the level of elements depends on the specific automorphism chosen. The four automorphisms are given below (the automorphism no. is just for convenience, it has no mathematical meaning):

There are a number of ways of seeing that there can be no further outer automorphism classes.

Combined structure of automorphism group: isomorphic to original group
Piecing together the given information, we see that the automorphism group is isomorphic to dihedral group:D8, with the outer automorphism $$\sigma_1$$ being an element of order four, and conjugation by $$x$$ being an element of order two that conjugates it to its inverse. Specifically, there is an isomorphism between the dihedral group of order eight and its automorphism group, given by $$a \mapsto \sigma_1$$ and $$x \mapsto c_x$$, where $$c_x$$ denotes conjugation by $$x$$.

Another way of seeing this is as follows: the automorphism group acts on the subset $$\{ x, ax, a^2x, a^3x \}$$ of the dihedral group, because this is the unique non-identity coset of the characteristic subgroup $$\langle a \rangle$$ and hence must be preserved by every automorphism. The action on this set is just like the usual permutation action of dihedral group:D8 as symmetries of a set of size four. The element $$\sigma_1$$ acts like the 4-cycle (rotation by $$\pi/2$$) and conjugation by $$x$$ acts like reflection about the diagonal joining $$x$$ and $$a^2x$$.

Although the dihedral group is a group isomorphic to its automorphism group, the homomorphism given by the action by conjugation is not an isomorphism. Rather, the homomorphism has kernel the center and its image is the inner automorphism group, which is the Klein four-subgroup discussed above.

Summary
Here is a summary of the various kinds of endomorphisms. Note that the first and second to last row are special in that the first row is about automorphisms and the second to last row is about the trivial endomorphism, and these are not elaborated upon further in the sections below:

Endomorphisms with kernel the center
There are 12 endomorphisms with kernel equal to the center of dihedral group:D8 $$\{e, a^2 \}$$. First, note that the quotient by the center is isomorphic to the Klein four-group. Next, note that there are two Klein four-subgroups of dihedral group:D8, so the endomorphism could choose either of them as its image. Finally, for any choice of image, there are as many choices for the actual mapping as the size of the automorphism group of the Klein four-group, which is $$GL(2,2)$$ or symmetric group:S3, and hence has order 6. Overall, we get $$2 \times 6 = 12$$ possibilities.

None of these endomorphisms are retractions. In a nilpotent group, an endomorphism with kernel equal to the center cannot be a retraction, because the center is not a direct factor; in fact, nilpotent implies center is normality-large.

Endomorphisms with kernel one of the Klein four-subgroups
There are two possibilities for this kernel, i.e., the two Klein four-subgroups of dihedral group:D8 $$\{ e, a^2, x, a^2x \}$$ and $$\{ e, a^2, ax, a^3x \}$$. The image could be any of the five copies of cyclic group:Z2, i.e., either the center of dihedral group:D8 $$\{ e, a^2 \}$$ or any of the four non-normal subgroups of dihedral group:D8, namely $$\{ e, x\}$$, $$\{ e, ax \}$$, $$\{ e, a^2x \}$$, $$\{ e, a^3x \}$$. For each choice of kernel and image, there is a unique endomorphism because cyclic group:Z2 has a trivial automorphism group. Thus, the total number of endomorphisms is $$2 \times 5 = 10$$.

Of all these endomorphisms, $$4$$ are retractions. Specifically, for each choice of Klein four-subgroup, there are two endomorphisms that send the non-identity coset to an element of order two outside the subgroup, and both of these are retractions. In total, there are $$2 \times 2 = 4$$ retractions.

Endomorphisms with kernel the cyclic group of order four
The kernel here is $$\{ e, a, a^2, a^3\}$$. The image could be any of the five copies of cyclic group:Z2, i.e., either the center of dihedral group:D8 $$\{ e, a^2 \}$$ or any of the four non-normal subgroups of dihedral group:D8, namely $$\{ e, x\}$$, $$\{ e, ax \}$$, $$\{ e, a^2x \}$$, $$\{ e, a^3x \}$$. For each choice of kernel and image, there is a unique endomorphism because cyclic group:Z2 has a trivial automorphism group. Thus, the total number of endomorphisms is $$1 \times 5 = 5$$.

Of these, $$4$$ are retractions, namely, the four endomorphisms whose image is a non-normal subgroup of the dihedral group.