Regular representations of $Z_2 \times Z_2$ and $Z_4$ and their decomposition I've tried to look for info about this but failed.
First off, what does a regular representation of $Z_4$ and $Z_2 \times Z_2$ look like? 
I know that a regular representation for a group $G$ is a homomorphism $b:G\to GL(G)$, over some field $\mathbb{K}$, which is given by $b(g) = f, f(x\in G) = gx$. (correct so far?)
Now the question implies that the regular representation on the above groups is 'special', I fail to find why. Also, how would one decompose them into dimension-$1$ representations?
 A: First of all, a representation of a group $G$ is a homomorphism $G\to \mathrm{GL}(KG)$, where $KG$ is the linear space
$$KG=\left\{\sum_{g\in G}\alpha_g g\mid \alpha_g\in K\right\}.$$
In particular, the set $G$ forms a basis for $KG$. In this basis, it is straightforward to compute the image of an element $h\in G$ inside $\mathrm{GL}(KG)$ since we have
$$h.\sum_g\alpha_g g=\sum_g\alpha_g hg=\sum_g \alpha_{h^{-1}g}g.$$
Indeed, this determines a $G\times G$ matrix with $(g_1,g_2)$-entry given by
$$\begin{cases}1&\mbox{if }g_1=h^{-1}g_2\\0&\mbox{otherwise}\end{cases}$$
For example, the group $C_4=\langle c\mid c^4=1\rangle\cong\mathbb{Z}_4$ acts on $KC_4$ (in the basis $\{1,c,c^2,c^3\}$) via the matrix
$$c\mapsto\begin{pmatrix}0&1&0&0\\0&0&1&0\\0&0&0&1\\1&0&0&0\end{pmatrix}.$$
The structure of such a representation is heavily dependent on the ground field $K$. In the example of $C_4$ above, the characteristic polynomial for the action of $c$ on $KC_4$ is $\chi_c(\lambda)=\lambda^4-1$. One needs to consider a couple of cases:


*

*If $\mathrm{char} K=2$, then $\chi_c(\lambda)=(\lambda-1)^4$, and it can be shown that $KC_4$ is an indecomposable module.

*If $\mathrm{char} K\neq 2$, then $KC_2$ decomposes as a direct sum of simple submodules. The nature of this decomposition depends on whether the polynomial $X^2+1$ splits in $K[X]$. If it does, then there is an element $i\in K$ of order 4, and the set $\{1+c+c^2+c^3,1-c+c^2-c^3,1+ic-c^2-ic^3, 1-ic-c^2+ic^3\}$ is a basis of eigenvectors for the action of $C_4$ on $KC_4$. If not, then there are two one dimensional submodules of $KC_4$ (spanned by the first two elements above) and a 2-dimensional irreducible submodule (spanned by $\{1-c^2, c-c^3\}$).
The case of $C_2\times C_2$ is studied similarly (but is a little easier). For example, the matrix for the generator $(c,1)$ of $C_2\times C_2$ on $K(C_2\times C_2)$ is given by
$$(c,1)\mapsto\begin{pmatrix}0&1&0&0\\1&0&0&0\\0&0&0&1\\0&0&1&0\end{pmatrix}.$$
The corresponding characteristic polynomial is $(\lambda^2-1)^2$. If $\mathrm{char} K\neq 2$, the module splits into four 1-dimensional submodules, but if the characteristic is 2, the module is not semisimple.
A: For $G = Z_4 = \{1, g, g^2,g^3\}$ written multiplicatively, let $V$ be the regular representation of $G$, basis $e_{g^i} : i = 0, 1, 2, 3$.  You can check that the linear transformation $T: V \rightarrow V$ given by $Tv = gv$ has four distinct eigenvalues: $1, -1, i, -i$.  Each eigenspace is $G$-invariant and one dimensional.  So if you calculate these eigenspaces, you can decompose $V$ as direct sum of the irreducible representations of $G$.  I'll try $G = Z_2 \times Z_2$ next.
