TITLE: Does every linear group admit a subgroup of dimension 1?
QUESTION [6 upvotes]: Suppose that $G$ is a linear group of positive dimension, defined over some field $k$. Is that true, that $G$ admits a (closed) one-dimensional subgroup?
I'm pretty much sure this is true in characteristic 0, or at least for $k=\mathbb{C}$. It seems that the main obstacle in positive characteristic is that there may not exist elements of infinite order.
EDIT1: As @Daniel Loughran pointed out in the answer below, one need at least to assume that $k$ is algebraically closed
(or it may not even be necessary, see the comments of @Marguax abaut $k$ being separably closed, or real closed in characteristic 0).
EDIT2: Since my first statement of the problem seems to be genereting a lot of confusion (after getting a comment from @Jim Humphreys I'm even a little ashamed), here is the final version of the question:
Suppose that $G$ is a linear algebraic group, of positive dimension, defined over an algebraically closed field $k$ (but of arbitrary characteristic). Is it true, that $G$ posses a one dimensional (closed) subgroup? (so in fact, either $\mathbb{G}_a$ or $\mathbb{G}_m$)

REPLY [10 votes]: Every positive dimensional linear algebraic group $G$ over an algebraically closed field has a one dimensional subgroup. 
Case 1 $G$ is reductive. In that case, $G$ contains a torus $T$. Since we are over an algebraically closed field, $T \cong \mathbb{G}_m^r$ for some $r>0$. In particular, $\mathbb{G}_m \subseteq T \subseteq G$.
Case 2 $G$ has a nontrivial unipotent radical $U$. In turn, $U$ has a nontrivial center $Z$. We have $Z \cong \mathbb{G}_a^s$ for some $s>0$ (not quite right, see Peter McNamara's correction below). In particular, $\mathbb{G}_a \subseteq Z \subseteq U \subseteq G$.

REPLY [7 votes]: Over an algebraically closed field, the answer is yes. I assume that by "linear group" you mean a smooth affine algebraic group scheme. Such a group has a filtration whose quotients (modulo finite group schemes) are successively a unipotent group, a torus, and a semisimple group, so it is only a question of checking each of these cases.