= Isogenous elliptic curves can have different rational point groups
{title2=$\#E(\mathbb F_q)=\#E^{\prime}(\mathbb F_q)\ \not\Rightarrow\ E(\mathbb F_q)\cong E^{\prime}(\mathbb F_q)$}
<Elliptic curves> isogenous over a <finite field> have the same number of rational points: the isogeny intertwines Frobenius, and multiplicativity of degree gives equal degrees for $1-\pi$. Their groups need not be isomorphic. Over $\mathbb F_7$, $y^2=x^3-x$ and $y^2=x^3+4x$ are two-isogenous and have groups $\mathbb Z/2\mathbb Z\times\mathbb Z/4\mathbb Z$ and $\mathbb Z/8\mathbb Z$, respectively.
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