Finding torsion subgroups of elliptic curves over finite fields.
Given $y^2=x^3+x+1$ over $F_3$ I need torsion subgroup of $E[3]$
$E[3]$ is either trivial or isomorphic to $\mathbb Z_3$
The points $(1,0),(-1,0),(0,0)$ are each of order $2$, so useless, but the point $(3,1)$ has order $4$ i.e. $4(3,1)=(\infty,\infty)$, Is there a possibility to combine this point with the others (or with itself, I cannot see it now) to get an element of order $3$ ?
To answer your specific question, no, you cannot "combine" points of order 2 and order 4 to create points of order 3. On the other hand, it's rather easy to find the points of order 3. Simply use the duplication formula to write $$x(2P)=x(P).$$ Clearing denominators will give you an equation to solve for $x(P)$. (In general, you'd get a quartic equation, but since you're looking for $p$-torsion in characteristic $p$, the degree will be lower.) If you get a constant, the curve is supersingular, and $E[3]=0$. If you get a non-constant equation, then $E[3]\cong\mathbb{Z}/3\mathbb{Z}$.