Group Theory

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The dimension of GL(n, R) is n^2 as a vector space (Vector spaces have dimensions)

I haven't got any immediate ideas for what it means with O(n). (Assuming O(n) is O(n, R) )

I think we're in the realms of Lie groups here, which is somewhat beyond basic group theory, and certainly beyond my current understanding. Wikipaedia has a couple of articles on General Linear Groups and Orthogonal Groups in relation to Lie group theory; may be of use - dunno.

I'm studying physics but missed out on the group theory course which seems to have been a mistake. I thought I understood the basics of group theory, but I'm having trouble understanding what is meant by the dimension of a group.

I keep seeing things like the dimesion of $GL(n, R)$ is $n^2$, $O(n)$ is $\frac{n(n-1)}{2}$ etc. I don't understand what this refers to. I would've thought that the dimension of the group was just n. I can't seem to find anything that explains this online :s

$\text{GL}(n, \mathbb{R})$, $\text{O}(n)$ are Lie groups, which basically means that they are simultaneously smooth manifolds and groups, and the group operations are smooth, in the sense that the maps $g \mapsto g^{-1}$ and $(g, h) \mapsto gh$ are smooth maps. The dimension of a Lie group is simply its dimension as a manifold. (Actually, this may not be well-defined if the group is not connected, e.g. $\text{O}(n)$. If we want to be precise, we can instead define the dimension of a Lie group to be the dimension of its Lie algebra, which is the tangent space at the identity element endowed with some additional structure.)

The only other context in which it makes sense to talk about the dimension of a group is when the group is also a vector space over some field, but these are rather boring groups. In the case where the group is a vector space over the reals, then, if it is finite-dimensional, it has a canonical smooth manifold structure which makes it into a Lie group (and a topological vector space) and the two definitions of dimension agree.

Oh dear. I see I have a lot of reading up to do...

Can you recommend any good places to start reading up about this? Bearing in mind that I'm a physicist not a mathematician?
Thanks

Any good physicist should know enough about differential geometry to understand basic Lie theory! But if you want to learn about basic Lie theory without such a background, perhaps Stillwell's Naive Lie Theory would be good.

Perhaps you're not meant to understand these claims at a deep level yet. At Cambridge at least these are 3rd/4th-year topics.