Group norm element ($N_G={\sum }_{g\in G}g$)

The group norm element in the group algebra $kG$ is $N_G={\sum }_{g\in G}g$. It spans the invariant subspace of the left regular group representation, and $N_G^2=|G|N_G$. If the characteristic divides $|G|$, a projection from $kG$ onto $k{N}_G$ would send $N_G$ to both itself and zero, proving that $kG$ is not a semisimple algebra. Its action on indecomposable modules is described by group norm element detects the trivial projective cover.