The general linear group of a vector space GL(V) is the group who's set is the set of all linear maps from V to V that are invertible (automorphisms). My question is, why is this a group? Surely the zero operator that sends all vectors in V to the zero vector is not invertible? But isn't it part of the definition of a group that an inverse exists for all elements? My book tells me that it is not a vector space because of the above, but I can't see how it is even a group! Can someone clear this up? Thanks.