3 year old DCG hitting

I have a new 3 year old dcg that has only been with us for a week. Today she was on a hitting spree, but only when I wasn't looking. She also kept accusing my older daughter (almost 4 years old) of hitting her first (making her hitting justified, I guess), but I know that isn't true because my daughter never ever hits and I trust her when she said she didn't do it. I think a lot of it has to do with the dcg wanting a toy and not wanting to wait to have her turn with it.

She is an only child and we are definitely having some issues with that and her possessiveness of things that don't belong to her. She will freak out of any other child even thinks of touching any of her possessions (like her hat or shoes) or she thinks that they are going to put their shoes where hers go. This is not her first time in daycare, but I think her last experience was a lot different than here.