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We give lower bounds for the rank of a symmetric fusion category incharacteristic $p\geq 5$ in terms of $p$. We prove that the second Adamsoperation $\psi_2$ is not the identity for any non-trivial symmetric fusioncategory, and that symmetric fusion categories satisfying$\psi_2^a=\psi_2^{a-1}$ for some positive integer $a$ are super Tannakian. Asan application, we classify all symmetric fusion categories of rank 3 and thoseof rank 4 with exactly two self dual simple objects.

Remarks on symmetric fusion categories of low rank in positive characteristic