TITLE: Dimension in Whitney's theorem
QUESTION [9 upvotes]: There are some interesting examples of (classes of) manifolds for which I suspect that the Whitney theorem can be strenghtened. For example it is known that every (smooth) $n$-dimensional manifold can be embedded into $\mathbb{R}^{2n-1}$ for example if $M$ is orientable or is not of dimension of the form $2^k$. In particular I'm interested what is the minimal number $emb(n)$ with the property that each $n$-dimensional manifold can be embedded into $R^{emb(n)}$ if we assume additionally that $M$ is:
a) simultaneously orientable and not of dimension of the form $2^k$,
b) a Lie group, and
c) the boundary of some other manifold?

REPLY [8 votes]: The following paper of Elmer Rees addresses precisely your question for Lie groups:
Rees, Elmer
Some embeddings of Lie groups in Euclidean space. 
Mathematika 18 (1971), 152–156. 
The main result is that if a compact connected Lie group $G$ of rank $\ell$ admits a faithful representation on $\mathbb{R}^k$, then $G$ admits an embedding into Euclidean space whose codimension is $k-\ell$. This improves on Whitney's Theorem when $k-\ell<\operatorname{dim}G$, which is the case for many of the classical families of Lie groups.