Yesterday, I started learning group theory.

Then I found a sample group theory definition like following (translated from japanese by me):

> 
> For example, consider the set V of all n-dimensional vectors. Addition
> and subtraction are defined in V. However, multiplication and division
> are not defined. Therefore, if we build a general theory about "sets
> in which addition and subtraction are defined", we can apply it to all
> things with similar properties. This is a "group".
> 
> 
> 

But I think it's weird. Because if we have two sample vectors like V1 and V2, it can be multiplied at least.

```
// 2-dimentional vector
>>> V1 = torch.randn(2,2)

// 3-dimentional vector
>>> V2 = torch.randn(2,2,2)
>>> V1*V2
tensor([[[-0.4868,  0.6598],
         [-0.2150, -0.4340]],

        [[-0.2066,  0.2782],
         [-0.0213, -1.5037]]])

```

I can't determine whether,
we define the "group" which have addition and subtraction not multiplication in this context
**OR**
originally set V of all n-dimensional vectors has addition and subtraction not multiplication.

Any suggestion ?

**EDIT**

Thank you all.
I misunderstood vector and matrix, and I didn't know multiplication on various situation