Definitions in a Theorem of Lang I'm trying to understand the following theorem due to Serge Lang (Algebraic Groups over Finite Fields, 1956, Theorem 2):

Let $p$ be a prime, and let $k$ be a finite field of $q=p^n$ elements.  Let $G$ be an algebraic group defined over $k$, and suppose $H/k$ is a homogeneous space over $G$.  Then $H$ has a rational point.

I've searched the paper to try to see what exactly Lang means by all of these terms, but I can't seem to find any answers.  I just have a few simple questions about some definitions.


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*By algebraic group defined over $k$, does Lang mean a group that is also a variety over $k$ where multiplication and inverse are morphisms of varieties?  If so, then $G$ must be finite because $k$ is finite.  Perhaps he means a variety over $\bar{k}$?

*By a homogeneous space $H/k$ over $G$, does Lang mean a variety $H$ over $k$ (or $\bar{k}$?) on which $G$ acts transitively and continuously?

*By rational point, does Lang mean some $h\in H$ such that the coordinates of $h$ in some embedding of $H$ into affine space over $\bar{k}$ all lie in $k$?  I guess here I'm assuming $H$ is an affine variety over $\bar{k}$.


Perhaps some of these questions don't make any sense, but the more I try to make them make sense, the more I realized I am just confused and need some straightening out.
 A: It sounds like you are confused about the definition of a variety over $k$ when $k$ is not algebraically closed. In particular, such an object is not faithfully described by its $k$-points; it contains more data, e.g. it has a notion of $L$-points for any algebraic extension $L$ of $k$. One way of describing this extra data is that a variety over $k$ is faithfully described by its $k_s$-points (where $k_s$ is the separable closure) together with the action of the absolute Galois group $\text{Gal}(k_s/k)$ on them (in the sense that this defines a functor from $k$-varieties to $\text{Gal}(k_s/k)$-sets which is faithful). "Rational point" here means $k$-point, or equivalently a fixed point for the Galois action. 
What level of formality you want to use to describe $k$-varieties depends on how much algebraic geometry you're comfortable with. Affine $k$-varieties are relatively easy to describe, in any case: this is the opposite category of the category of finitely-generated integral (or reduced, depending on your preferences) commutative $k$-algebras. 
A: I've done some reading out of Humphreys' "Linear Algebraic Groups," and I believe Lang is referring to definitions found in section 34 entitled "Fields of Definition."  Here is what I've found.  We will work purely in the affine case.
Let $k\hookrightarrow K$ be a map of fields, and suppose $X\subset\mathbb{A}^n_K$ is a closed set.  $X$ is said to be defined over k if $\mathscr{I}(X)$ is generated by elements in $k[x_1,\ldots,x_n]$.  If $X\subset\mathbb{A}^n_K$ and $Y\subset\mathbb{A}^m_K$ are two closed sets defined over $k$, then a map $\varphi:X\rightarrow Y$ is a $k$-morphism if the coordinate functions all lie in $k[x_1,\ldots,x_n]$.  Then, we say an algebraic group $G$ is defined over $k$ if it is defined over $k$ as a closed set, and if multiplication and inversion are $k$-morphisms.
We say an algebraic group $G$ defined over $k$ acts $k$-morphically on a closed set $X$ (also defined over $k$) if the action $\varphi:G\times X\rightarrow X$ is a $k$-morphism
A homogeneous space $H/k$ is a closed set $H\subset\mathbb{A}^n_K$ defined over $k$ together with a transitive $k$-morphic action of $G$ on $H$.
Finally, if $X\subset\mathbb{A}^n_K$ is a closed set defined over $k$, then the $k$-rational points of $X$ is the set $X(k)=X\cap \mathbb{A}^n_k\subset\mathbb{A}^n_K$.
Considering rings of regular functions, all of this can be defined intrinsically, without reference to a specific embedding, and can be extended to arbitrary varieties as well by affine open coverings.
With all of this in mind, I think that Lang is actually considering $G$ and $H$ as varieties over some larger field $K$ of characteristic $p$ containing $k$ (for example,  $K=\bar{k}$), and requiring that they be defined over $k$ as described above.
