In the high energy physics literature and in some discussions on group theory for physics, the tensor product $\otimes$ of groups is sometimes mentioned. For example, the gauge group of the standard model is sometimes said to be $SU(3)\otimes SU(2)\otimes U(1)$. Is this "tensor product" really just a cartesian product/direct product, or is it something else?
Is there any difference between $SU(3)\otimes SU(2)\otimes U(1)$ and $SU(3)\times SU(2)\times U(1)$?
Note: I previously asked this question but it was closed for being more about "history of science" (?) than physics. I hope the rephrased version here is more on-topic.
