I am starting to study group algebras and I am stuck on the following problem. The first part is easy, but I copy it in case it helps to prove the second part. This exercise is taken from Representations of groups by Lux and Pahlings.
Suppose $K$ is a field with char $\neq 2$ containing a primitive $4$th root of unity $i$, and let $\langle g \rangle = C_4$. Put $a = \frac{1+i}{2}g+\frac{1-i}{2}g^3 \in KC_4$ and $b = \frac{1-i}{2}g+\frac{1+i}{2}g^3 \in KC_4.$
(a) Show that $\{1,g^2,a, b\} \subseteq KC_4$ is a subgroup of the unit group of $KC_4$ isomorphic to $V_4 \cong C_2 \times C_2$. (easy part)
(b) Show that $KC_4 \cong KV_4$ as algebras over $K$.
Here is my problem: I can't find an isomorphism between $KC_4$ and $KV_4$. Actually, I don't understand a priori how they could be isomorphic since $C_4 \ncong V_4$ as groups.
As a natural second question, - since I guess that there should actually exist an isomorphism after all - I wonder: are the hypotheses "char$K\neq 2$ and there exists a primitive 4th root of unity in $K$" necessary in order to obtain part (b)?
I think my problem is partly conceptual: I don't really have any intuition on how to work with group algebras. Could you give me hints or help me to get a clearer view on this topic?