Let $G$ be a group of order $n$ where field is real or complex. Does there exist a faithful $FG$-module of dimension $n$ (in other words a representation of degree $n$) other than the regular $FG$– module (or representation)?
I think it should be no, as every such representation will be isomorphic to regular one.
Best Answer
They can exist. For instance, consider the Klein four group: the regular rep decomposes as a direct sum of all four 1D irreps, but we can construct another faithful 4D rep by direct summing any four 1D reps at least two of which are distinct nontrivial. Indeed, you can take the direct sum of two distinct nontrivial permutation 2D reps.