On the automorphism group of a closed G2-structure

We study the automorphism group of a compact 7-manifold M endowed with a closed non-parallel G2-structure, showing that its identity component is abelian with dimension bounded by min{6,b2(M)}⁠. This implies the non-existence of compact homogeneous manifolds endowed with an invariant closed non-parallel G2-structure. We also discuss some relevant examples.