### group algebra

Can anyone help in writing ~~a ~~code to find the list of idempotent and primitive elements of a group ~~algebra, the ~~algebra?

The examples goes like this. Let ~~p ~~$p$ be an odd prime such that $\bar2$ generates $U(Z_{p^2})$ and ~~G=(ideal ~~let $G =(\text{ideal generated by ~~a)∗ (ideal ~~}a) ∗ (\text{ideal generated by ~~b) ~~}b)$ an abelian group, with $o(a)=p^2$ and ~~$o(b)=p$. ~~$o(b)=p$.

Then $F_2G$ has four inequivalent minimal codes, namely, the ones generated by the ~~idempotents $e_0= \hat{G}$, $e_1=\hat{b}−\hat{(ideal ~~idempotents:

$$e_0 = \hat{G}$$
$$e_1=\hat{b}−\widehat{(\text{ideal generated by ~~'a')∗ (ideal ~~}a)∗ (\text{ideal generated by ~~'b')}$(note: its a big hat), $e_2=\hat{a−G}$ and $e_3=\hat{(ideal ~~}b)}$$
$$e_2=\widehat{a−G}$$
$$e_3=\widehat{(\text{ideal generated by ~~'a_p')∗ (ideal ~~}a_p)∗ (\text{ideal generated by ~~'b')}−\hat{G}$.~~}b)}−\hat{G}$$