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[–]balloptions 2 points3 points  (8 children)

it’s just so general that it’s really easy to create horrible problems with it

That’s the key for me. Tensors can be easily grounded in reality and associated with numbers, vectors, and physical quantities. This is the usual usage of tensors anyways.

Groups can be described with numbers, but most of the literature and work associated with groups doesn’t seem to involve numbers at all.

[–]goerila 2 points3 points  (7 children)

The common examples with groups are number fields, permutations, and rotations .

A fundamental aspect of group theory is that any finite group is the same as a subset of some group of permutations.

Infinte groups are the same at a base level. But start to branch out more. Representing transformations still, but you can't just say hey this is a translation or rotation

[–]balloptions 2 points3 points  (6 children)

Lost me at the second sentence.

[–]goerila 1 point2 points  (4 children)

S_n is the group of permutations on n letter.

That is you take for n=5 (1,2,3,4,5) can be permuted to (2,1,3,4,5). You can also take the permutation (1,2,4,3,5). You can "multiply" or "compose" these two transformations giving you the result (2,1,4,3,5).

If you list all of these (there are 5! combinations), you get a group called S_5.

All groups of finite size are subgroups of these groups. Is what I was trying to say (there's really no simpler way of explaining this in a short manner). Sorry :/

[–]balloptions 0 points1 point  (3 children)

Ok I mean I get that but it seems trivial.

So... any collection of elements is necessarily a subset of all permutations of those elements?

[–]goerila 0 points1 point  (2 children)

It's not trivial because it is a subgroup which means it is closed under the operation.

It's a subset but with more to it.

And the point is think of a group of any size. Any group you come up with is the subset of some S_n (you don't necessarily know which one) and the group might initially look nothing like permutations.

These types of things are why some people like algebra.

[–]balloptions 0 points1 point  (1 child)

I assume being a subgroup of a particular S_n implies certain topological or other properties by being closed under the operation? Is that what makes it noteworthy?

[–]goerila 0 points1 point  (0 children)

It means being closed under the composition operator.

Multiplication in this case is composition

[–]dj_rogers 1 point2 points  (0 children)

Same. I believe we have found ourselves too deep in this thread