The confusion comes from a geometry text in which the concept of dilatation is given as follows (I use image for the purpose of a faithful reproduction of the problem):
This proposition and definition appears in a chapter on affine spaces and morphisms. The space is denoted as $(X,\vec X, \Phi)$ where $X$ is the affine space, $\vec X$ the underlying vector space and $\Phi$ the group action. $T(X)$ is the set of all translations $\{\Phi(\vec\xi)=t_{\vec\xi}|\vec\xi\in\vec X\}$. ${\rm{GL}}(\vec X)$ is the linear group of $\vec X$. $K$ is the scalar field and $K^*=K-\{0\}$. ${\rm{GA}}(X)$ is the affine group of $X$. $L$ is the homomorphism ${\rm{GA}}(X)\ni f\to\vec f\in{\rm{GL}}(\vec X)$. $H_{a,\lambda}$ denotes homothety of center $a$ and ratio $\lambda$: $x\to a+\lambda\vec{ax}$. Some facts follows: 1) $\ker L=T(X)$. 2) Let $f\in{\rm{GA}}(X)$ be such that $\vec f=\lambda{\rm{Id}}_{\vec X}$ where $\lambda\in K,\lambda\ne0,1$, it can be shown that there is a unique $a$ such that $f=H_{a,\lambda}$. I write the above here to avoid inconsistency on terminology. Now comes my question.
In my understanding of group homomorphism, the inverse image of a point in range is a coset with regard to the kernel. Since the operation in ${\rm{GA}}(X)$ is composition and the kernel is $T(X)$, the elements in coset has the form $H_{a,\lambda}\circ t_{\vec\xi}$ and I have shown for verification that $\overrightarrow{H_{a,\lambda}\circ t_{\vec\xi}}$ is indeed $\lambda{\rm{Id}}_{\vec X}$. So the inverse image ${\rm{Dil}}(X)$ of set $K^*{\rm{Id}}_{\vec X}$ should be the union of all aforementioned cosets, i.e., the "product" (actually the semidirect product) of all $H_{a,\lambda}$ for $a\in X$ and $\lambda\in K^*$(including $1$) with $T(X)$, instead of the "union" of them as I underlined in the text in red. Each coset contains a unique homothety $H_{a,\lambda}$ while all other elements in the coset (that is, dilatations) are a composition of the homothety with a translation. Now my question is simple: can you please confirm my understanding (and hence that the text is incorrect in using the word "union")?
Perhaps another lemma in the text related to dilatations may be helpful in clarifying what the elements of ${\rm{Dil}}(X)$ is:
In the above lemma, the author first says $f$ is a dilatation. Then he goes on to assume $f\in T(X)$ in the first case, a translation, and then $f$ is a homothety of center $0$ in the second case. This is another place where I got confused. From common writing rules, $f\in T(X)$ and $f$ is a homothety should be two concrete examples of $f$ being a dilatation. Am I right? If I'm correct, it can be inferred that dilatation can take a form of either translation or homothety. If the author means translations and homotheties are the only dilatations, this is consistent with his definition by means of "union". But are there any other form of dilatations, those in my point of view in the coset, that are not translations or homotheties and therefore do not fall into any case in the lemma but that have its own result (e.g., $f(b)$ is on line $D^\prime$). I would greatly appreciate it if anyone having background of college geometry could help me clarify what are the dilatations. Please let me know if you have any difficulty in understanding my question (say, what $\vec f$ is). Thank you.
For reference purposes, the book in question is M. Berger's Geometry I; 2.3.3.12 is on p. 40.
You seem to have answered your own question: Indeed, any dilatation is either a homothety (with ratio not equal to $1$) xor it is a translation.
Let me recap aspects of the theory in my own slightly different notation (the details can be found in Berger's book). For $\mathbb{K}$ a field, an affine space over $\mathbb{K}$ is a triple $(X,V,a_\bullet)$, where $X$ is a set, $V$ is a vector space over $\mathbb{K}$ and $a_\bullet: (V,+)\to \operatorname{Bij}(X)$ is a faithful and transitive (hence also free) group action of $V$ considered as an abelian group. By the freeness of the action we have a structure or transition map or cocycle $\Theta:X\times X\to V$ that takes a pair $(x_1,x_2)$ and sends it to the unique solution $v\in V$ to the equation $a_v(x_1)=x_2$. We can thus define the difference of two points of $X$:
$$-x_1+x_2 := \Theta(x_1,x_2) =\mbox{ the unique vector that takes $x_1$ to $x_2$}.$$
(I will avoid using this abbreviation, though it's very useful as an intuition pump.)
Using the transition cocycle we have isomorphisms of $(V,+)$-sets:
$$\forall x_0\in X: \Theta_{x_0}:=\Theta(x_0,\bullet): X\xrightarrow{\cong} V, x\mapsto \Theta(x_0,x),$$
and if we fix any $x_0$ in $X$ we can use $\Theta_{x_0}$ to endow $X$ with a $\mathbb{K}$-vector space structure. If we do so this vector space is denoted by $X_{x_0}$. This means explicitly that in $X_{x_0}$ the vector space operations work like so:
$$cx_1+x_2 = \Theta_{x_0}^{-1} (c\;\Theta_{x_0}(x_1)+\Theta_{x_0}(x_2)), \quad\quad (\dagger)$$
and that the zero vector of $X_{x_0}$ is $x_0$. Observe that this is the first instance when scaling comes into the picture.
If $(X,V,a_\bullet)$ and $(Y,W,b_\bullet)$ are two $\mathbb{K}$-affine spaces, a morphism of $\mathbb{K}$-affine spaces $f:(X,V,a_\bullet)\to (Y,W,b_\bullet)$ from the former to the latter is by definition a function $f:X\to Y$ such that
$$\exists x_0\in X: f_{x_0}:=H_{f(x_0)}\circ f \circ \Theta_{x_0}^{-1} : V\to W \mbox{ is $\mathbb{K}-$linear}. \quad\quad(\ast)$$
($H$ is supposed to be capital eta, which I used for the transition cocycle of the latter affine space.)
Changing the basepoint has the following effect for any function $f$ from $X$ to $Y$ whatever:
$$\forall x_1,x_2\in X, \forall v\in V: f_{x_2}(v) = f_{x_1}(\Theta(x_1,x_2)+v)-f_{x_1}(\Theta(x_1,x_2)).$$
This means in particular that the test ($\ast$) for a function $f:X\to Y$ to be affine is independent of the basepoint $x_0$ chosen; which in turn means that if for some $x_0\in X$: $f_{x_0}$ is linear, then $f_x$ is linear for any $x\in X$, since they are all the same linear map, which is denoted by $L(f)$.
As expected the set of all affine automorphisms of $X$ form a group $GA(X)$ and $L: GA(X)\to GL(V)$ hence defines a group homomorphism, which we may call the linear part homomorphism. Recall that the kernel $\ker(L)$ of $L$ in this situation is the group of all affine automorphisms of $X$ whose linear part is $\operatorname{id}_V$. Using the pointwise realization ($\ast$) at an arbitrary point $x_0\in X$ we see that
$$\forall f\in \ker(L): f=\Theta_{f(x_0)}^{-1}\circ \operatorname{id}_V \circ \Theta_{x_0}=\Theta_{f(x_0)}^{-1}\circ \Theta_{x_0} = a_{\Theta(x_0,f(x_0))}.$$
In other words the kernel of $L$ is precisely the image of $a_\bullet$ in $\operatorname{Bij}(X)$:
$$\operatorname{im}(a_\bullet) = \{a_v| v\in V\} = \ker(L)=: T(X).$$
Another important definition is that of a homothety. For $x_0\in X$ and $\lambda \in \mathbb{K}^\times $ (which is the set of all units in $\mathbb{K}$), the homothety $h_{x_0,\lambda}$ with center $x_0$ and ratio $\lambda$ is by definition the function
$$h_{x_0,\lambda}:X\to X, x\mapsto a_{\lambda\Theta(x_0,x)}(x_0).$$
Three observations are in order:
By Obs. 3, if we have a fixed ratio $\lambda$ for a homothety, a point in $X$ completely determines the homothety (as the one that fixes said point). Let us denote by $\operatorname{Htt}(X)$ the set of all homotheties of $X$ and by $\operatorname{Htt}_\lambda(X)$ ($\lambda\in \mathbb{K}^\times$) the set of all homotheties of $X$ with ratio $\lambda$. Thus
\begin{align*} \operatorname{Htt}(X) &= \biguplus_{\lambda\in\mathbb{K}^\times} \operatorname{Htt}_\lambda(X) = \left(\biguplus_{\lambda\in\mathbb{K}^\times\setminus1} \operatorname{Htt}_\lambda(X)\right) \uplus \operatorname{Htt}_1(X)\\ &= \left(\biguplus_{\lambda\in\mathbb{K}^\times\setminus1} \operatorname{Htt}_\lambda(X)\right) \uplus \{\operatorname{id}_X\}. \end{align*}
Here $\uplus$ stands for (internal) disjoint union and the last equality is because of Obs. 1.
By Obs. 2, for any $\lambda\in\mathbb{K}^\times$, $\operatorname{Htt}_\lambda(X)\subseteq L^{-1}(\lambda \operatorname{id}_V)$. Now the crucial lemma is:
Lemma: Let $\lambda\in\mathbb{K}^\times\setminus 1$. Then the inclusion map $\operatorname{Htt}_\lambda(X)\to L^{-1}(\lambda \operatorname{id}_V)$ is surjective whose inverse is given by:
$$L^{-1}(\lambda \operatorname{id}_V) \to \operatorname{Htt}_\lambda(X), f\mapsto h_{x_f,\lambda},$$
where
$$x_f:=\Theta_{x_1}^{-1}\left(\dfrac{1}{1-\lambda}\Theta_{x_1}(f(x_1))\right)$$
is the unique fixed point of $f$ that is independent of the basepoint $x_1\in X$.
(This is Prop.2.3.3.10 on p. 40 of Berger's book; there it looks much more pleasant because the transition cocycle is suppressed. Observe that the whole point of introducing $X_{x_1}$ actually is executing this suppression systematically; see ($\dagger$).)
By definition a dilatation of $X$ is an affine automorphism whose linear part is of the form $\lambda \operatorname{id}_V$ for some $\lambda\in\mathbb{K}^\times$ and the set of all dilatations of $X$ is denoted by $\operatorname{Dil}(X)$. Putting everything together we finally have:
\begin{align*} \operatorname{Dil}(X) &= L^{-1}(\mathbb{K}^\times \operatorname{id}_V) = \biguplus_{\lambda\in\mathbb{K}^\times} L^{-1}(\lambda \operatorname{id}_V) = \left(\biguplus_{\lambda\in\mathbb{K}^\times\setminus1} L^{-1}(\lambda \operatorname{id}_V)\right) \uplus L^{-1}(1\operatorname{id}_V)\\ &= \left(\biguplus_{\lambda\in\mathbb{K}^\times\setminus1} \operatorname{Htt}_\lambda(X) \right)\uplus \ker(L) = \left(\operatorname{Htt}(X)\setminus\{\operatorname{id}_V\} \right)\uplus T(X). \end{align*}
Thus we see that indeed a dilatation is either a homothety or it's a translation. If it's both; it it's trivial.
There is a heuristic picture that I think is helpful: Take $\mathbb{K}=\mathbb{R}$. We have the real line without its origin, parametrizing the center of the general linear group. Above it floats the set of all dilatations of $X$. $L$ projects down. Above $1\in\mathbb{R}^\times$ we see $T(X)$, which is nothing but $V$. Above any other point $\lambda\in\mathbb{R}^\times$ we see $\operatorname{Htt}_\lambda(X)$, which are parametrized by their fixed points, so we see a copy of $X$. Of course by the definition of an affine space one can also consider $X$ to be a copy of $V$.
On the other hand you are right that the dilatations can be considered as a semidirect product by way of considering homotheties. Indeed we have a split short exact sequence
$$1\to T(X)\to \operatorname{Dil}(X)\xrightarrow{L} \mathbb{K}^\times \to 1,$$
with $h_{x_0,\bullet}: \mathbb{K}^\times \to \operatorname{Dil}(X), \lambda\mapsto h_{x_0,\lambda}$ providing the splitting ($x_0$ an arbitrary basepoint) (See 2.3.3.11 on p. 40 of Berger's book). Thus $\operatorname{Dil}(X)\cong T(X)\rtimes \mathbb{K}^\times$, just as $GA(X)\cong T(X)\rtimes GL(V)$.