Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a Frobenius morphism of $G$, which defines a $\mathbb{F}_q$-structure $G(\mathbb{F}_q)$ on G .
QUESTION: Is the order of any element in $G$ finite?
If the answer is YES, I feel very surprised. But it seems that I can prove it as following.
Denote by $\bar{\mathbb{F}}_q[G]$ the ring of regular functions of $G$. Because $G$ has a $\mathbb{F}_q$-structure defined by $F$, $\bar{\mathbb{F}}_q[G]=\mathbb{F}_q[G]\otimes_{\mathbb{F}_q}\bar{\mathbb{F}}_q$. Let $a_i$ for $1\leq i\leq k$ be generators of $\mathbb{F}_q[G]$. Because there exsits some $n\in\mathbb{Z}^+$ such that $a_i(g)\in\mathbb{F}_{q^n}$ for all $i$, it follows that $F^n(g)=g$. On the other hand, $F^n$ is a Frobenius morphism attached to an $F_{q^n}$-structure on $G$, and hence the set of fixed points $G^{F^n}$ forms a finite group. Now, $g$ lies in the finite group $G^{F^n}$ and hence has a finite order.
If this is true, is there any reference book?
The underlying question is very easy, but maybe some perspective here is helpful, though the question is poorly phrased (as the down votes suggest, not due to me).
Historically, there was quite a bit of experimentation with the foundations of algebraic geometry. For instance, in Weil's Foundations one had to work in a very large field with lots of transcendental elements available in order to get "generic points". Chevalley developed a different but also unsatisfactory approach. Eventually the introduction of schemes (including group schemes) brought more clarity to the foundations.
However, when Borel lectured in the late 1960s on linear (= affine) algebraic groups in arbitrary characteristic, he evolved a sort of hybrid language to avoid as much formality as found in SGA3. This approach, with assistance from Bass, became a set of published notes and eventually an expanded Springer GTM volume. Here one still works mainly over fields (rather than more general rings), but arbitrary fields of definition and arbitrary algebraically closed fields are admitted. Here your $G$ is shorthand for the group of rational points over an algebraic closure.
Over a finite field, the key point is that an algebraic closure is countable and can be viewed as the union (or direct limit) of finite fields. Thus all nonzero elements of the multiplicative group have finite order. This propagates to general linear groups over an algebraic closure, as Jay points out.
I don't recall how Geck discusses all of this, but to be very concrete one might find it illuminating to look at the Jordan decomposition in $G$: here the unipotent elements are just those of $p$-power (hence finite) order, while the eigenvalues of semisimple elements lie in the multiplicative group of the big field (hence those elements too have finite order).