Let $G$ be an algebraic group. Given an algebraic subvariety $X\subseteq G$, is there any way simple criterion to determine whether $X$ has a group structure or not?
For example, when $X$ and $G$ are affine, then if $\mathcal{A}(G)$ and $\mathcal{A}(X)$ are the coordinate rings of $G$ and $X$, we know that $G$ being a group variety is equivalent to $\mathcal{A}(G)$ having a Hopf algebra structure. Since $\mathcal{A}(X)$ is a quotient of $\mathcal{A}(G)$, doesn't it immediately imply that $\mathcal{A}(X)$ also have a Hopf algebra structure?
An obvious immediate obstruction to $X$ having a group structure is if $X$ does not contain the identity. I assume this should somehow also be data one could read from the Hopf algebra structure.
Thanks in advance!
As in the comment by LT1918, the issue is that not all ideals $I$ in a hopf algebra $A$ are Hopf ideals, namely the comultiplication on $A$, $\Delta : A \to A\otimes_k A$ ($k$ being the ground field), need not induce comultiplication on the quotient $$ \overline{\Delta} : A/I \to A/I \otimes_k A/I; $$ for instance, setting $G=\mathbb{G_a}/k$ so $A=k[X], \Delta:X \mapsto X\otimes 1 + 1\otimes X$, implies that for example the ideal $I= (X(X-1))\subseteq k[X]$ isn't Hopf, since the composition $$ \Delta : k[X] \to k[X]\otimes_k k[X] \xrightarrow{\pi\otimes_k\pi} k[X]/I \otimes_k k[X]/I $$ doesn't factor through $\pi$, as $$ \Delta(X(X-1)) = X^2\otimes 1 + 1\otimes X^2 +2X\otimes X - 1\otimes 1 - X\otimes 1 = (X^2 - X)\otimes 1 + 1\otimes (X^2 - 1) + 2X\otimes X \neq 0 \text{ in }k[X]/I\otimes k[X]/I. $$ This is clear because I chose an ideal which doesn't define a subgroup of $\mathbb{G}_a$ of course. The wiki article has more than what you'd need on Hopf ideals I think.
Unless I'm mistaken, I think your question is on possible group structures of a subvariety which aren't necessarily induced by that of $G$, which I think is about as general as asking which varieties are algebraic groups... I don't know whether this has a satisfying answer tbh.
Sorry for the hazy answer and I hope this helps! :)