A Note on Linear Complementary Pairs of Group Codes

We give a short and elementary proof of the fact that for a linear complementary pair $(C,D)$, where $C$ and $D$ are $2$-sided ideals in a group algebra, $D$ is uniquely determined by $C$ and the dual code $D^\perp$ is permutation equivalent to $C$. This includes earlier results of Carlet et al. and G\"uneri et al. on nD cyclic codes which have been proved by subtle and lengthy calculations in the space of polynomials.