This is a basic question as I am just starting to learn how to work with tensors. I wanted to know whether a sequence of tensors fully contracted together commutes i.e.:
$$\tag{1} A^a_{\ b}B^b_{\ c} C^c_{\ a} = A^a_{\ b}C^c_{\ a}B^b_{\ c}, $$ where the Einstein summation convention is assumed through out. I imagine this is true as both addition and multiplication are commutative. Furthermore however, if indeed $$ A^a_{\ b}B^b_{\ c} C^c_{\ a} = A^a_{\ b}G^c_{\ a}B^b_{\ c}, $$ could one then conclude
$$\tag{2} C^c_{\ a} = G^c_{\ a},$$ i.e. $C=G$ as tensors or would this only hold up to some commutation relation?